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\vglue 30pt

\centerline{\bf STABLE INTERSECTIONS OF REGULAR CANTOR SETS}
\centerline{\bf WITH LARGE HAUSDORFF DIMENSIONS}

\vglue .3in

\centerline{Carlos Gustavo T. de A. Moreira \& Jean-Christophe Yoccoz}

\vglue .2in

\footnotetext" "{This work has been partially supported by 
Pronex-Dynamical Systems, Brasil.}


\vglue .5in

\noindent{\bf Abstract}: In this paper we prove a conjecture by J. Palis according to which the arithmetic difference of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval. More precisely, we prove that if the sum of the Hausdorff dimensions of two regular Cantor sets is bigger than one then, in almost all cases, there are translations of them stably having intersection with positive Hausdorff dimension.

\vglue .5in

\noindent{\bf Introduction}

\vglue .2in

Regular Cantor sets on the line play a fundamental role in dynamical systems and notably also in some problems in number theory. They are defined by expansive maps and have some kind of self-similarity property: small parts of them are diffeomorphic to big parts with uniformly bounded distortion (see precise definition in section I). In both settings, dynamics and number theory, a key question is whether the arithmetic difference of two such sets contains an interval 
when the sum of their Hausdorff dimensions is bigger than one. Some background on regular Cantor sets which are relevant to our work can be found in [PT2] and [Su].

From the dynamics side, in 1983, J. Palis and F. Takens ([PT], [PT1])
proved a theorem about homoclinic bifurcations associated to a basic set that assures full density of hyperbolicity in the parameter family provided that the Hausdorff dimension of the basic set is smaller than one. A central fact used in the proof is that if $K_1$ and $K_2$ are regular Cantor sets on the real line such that the sum of their Hausdorff dimensions is smaller than one, then $K_1-K_2 =
\{x-y \mid x \in K_1, y \in K_2\}$ (the arithmetic difference between $K_1$ and $K_2$) is a set of zero Lebesgue measure (indeed of Hausdorff dimension smaller than 1). In the same year, looking for some kind of converse of this result Palis conjectured (see [P], [P1]) that for generic pairs of regular Cantor sets $(K_1, K_2)$ of the real line either $K_1- K_2$ has zero measure or else it contains an interval (the last statement should correspond in homoclinic bifurcations to open sets of tangencies). A slightly stronger startement is that,
 if $K_1$ and $K_2$ are generic regular Cantor sets and the sum of their Hausdorff dimensions is bigger than 1, then $K_1-K_2$ contains intervals. 

Another motivation for the conjecture was Newhouse's work in the seventies ([N1], [N2], [N3]), when he introduced the concept of thickness of a regular Cantor set. This allowed him to exhibit examples of pairs of Cantor sets whose arithmetic difference stably is an interval, and to show that after a homoclinic bifurcation there are always open sets of stable tangencies in the parameter line. Is to be noted that in general (as in the [PT1] case above) these sets can have zero density.

Some previous works related to Palis' conjecture are [S], [BPV], [BPMV], [L], [MO] and [PY].

An earlier and totally independent development had taken place in number theory. In 1947, M. Hall ([H]) proved that any real number can be written as the sum of two numbers whose continued fractions  coefficients (of positive index) are at most 4. More precisely, $C(4)$
 is the regular subset formed of such numbers in $[0,1]$, one has $C(4)+C(4) = [\sqrt2 -1, 4(\sqrt 2 -1)]$. It follows 
that the Lagrange spectrum $L = \{\underset{\Sb p,q\to\infty\\ p,q\in{\na}\endSb}\to{\limsup}\,(q|q\al-p|)^{-1}$, $\al \in \re\backslash\bq\}$ contains a whole half-line (for instance $[6, +\infty)$). In 1975, G. Freiman ([F], see also [CF])proved some
difficult results showing that the arithmetic sum of certain (regular)
Cantor sets related to continued fractions contain intervals, and used them to determine the precise beginning of Hall's ray (the biggest half-line contained in $L$), which is 
$$
\frac{2221564096 + 283748\sqrt{462}}{491993569} \cong 4,52782956616\dots\,.
$$

In 1993, the concept of stable intersection of two regular
Cantor sets was introduced (see [M]): two Cantor sets $K_1$ and $K_2$ have stable intersection if there is a neighbourhood $V$ of $(K_1,K_2)$ in the set of pairs of $C^{1+}$-regular Cantor sets (see section 2 for a definition of the topology) such that $(\widetilde
K_1, \widetilde K_2) \in V \Rightarrow \widetilde K_1 \cap \widetilde K_2 \ne \emptyset$. In the same paper conditions based on
renormalizations were introduced to ensure stable intersections, and applications of stable intersections to homoclinic bifurcations were obtained, namely positive density of stable tangencies at the initial bifurcation point. It was also shown that this last phenomena can coexist with positive density of hyperbolicity in a
persistent way. Besides, the following question is posed:
Does there exist a dense (and automatically open) set $U \subset \Om^\infty =\{(K_1,K_2)$,\,\,$K_1$ and $K_2$\,\,\,$C^k$-regular Cantor sets $\mid HD(K_1) + HD(K_2) > 1\}$ such that $(K_1,K_2) \in U \Rightarrow \exists\,t\in\re$ st. $(K_1,K_2+t)$ has stable intersection?

A positive answer to this question implies a strong version of Palis'
conjecture. In the present work we will provided an affirmative answer to it, by proving (see section \S 2.4) the following

\vglue .2in

\noindent{\bf Theorem}: There is an open and dense set $U \subset \Om^\infty$ such that if $(K_1,K_2) \in U$, then $I_s(K_1,K_2)$ is dense in $K_1-K_2$ and $HD((K_1-K_2)\backslash I_s(K_1,K_2)) < 1$, where $I_s(K_1,K_2) := \{t \in \re \mid K_1$ and $(K_2+t)$ have stable intersection$\}$.

\

We will also prove that the same result works if we replace stable intersection
by $d$-stable intersection, which is defined by asking that any pair $(\widetilde K_1, \widetilde K_2)$ in some neighbourhood of $(K_1,K_2)$  satisfies $HD(\widetilde K_1 \cap \widetilde K_2) \ge d$: most pairs of Cantor sets $(K_1,K_2) \in \Om^\infty$ have $d$-stable intersection for any $d < HD(K_1) + HD(K_1)-1$ (see section \S 2.6).

Moreover, we will show that the open set $U$ mentioned in the theorem above is very large in $\Om^\infty$ in the sense that generic $n$-parameter families in $\Om^\infty$ are actually contained in $U$ (see section \S 3.9.1).

In a forthcoming paper, we prove the following fact concerning
generic homoclinic bifurcations associated to two dimensional 
saddle-type hyperbolic sets (horseshoes) with Hausdorff dimension bigger than one: it yield open sets of stable tangencies in the parameter line with positive density at the initial bifurcation value. Moreover, the union of this set with the hyperbolicity set in the parameter line generically has full density at the initial bifurcation value. This extends a theorem by Palis and Yoccoz ([PY]).

\vglue .3in

\noindent{\bf \S1.\quad Regular Cantor sets: the setting}.

\vglue .2in

\noindent{\bf 1.1}\quad Let $\AA$ be a finite alphabet, $\B$ a subset of $\AA^2$, and $\Sigma$ the subshift of finite type of $\AA^{\bz}$ with allowed transitions $\B$.

We will always assume that $\Sigma$ is topologically mixing, and that every letter in $\AA$ occurs in $\Sigma$.

An {\it expansive map of type\/} $\Sigma$ is a map $g$ with the following properties:

\item{(i)} the domain of $g$ is a disjoint union
$\underset{\B}\to{\bigcup}\,I(a,b)$, where, for each $(a,b)$,\,\, $I(a,b)$ is a compact subinterval of $I(a) := [0,1]\times\{a\}$;
\item{(ii)} for each $(a,b) \in \B$, the restriction of $g$ to $I(a,b)$ is a smooth diffeomorphism onto $I(b)$ satisfying $|Dg(t)| > 1$ for all $t$.

The {\it regular Cantor set\/} associated to $g$ is the maximal invariant set
$$
K = \bigcap_{n\ge0} g^{-n}\bigg(\bigcup_{\B} I(a,b)\bigg).
$$

Let $\Sigma^+$ be the unilateral subshift associated to $\Sigma$. There exists a unique homeomorphism $h\colon \Sigma^+ \to K$ such that
$$
\align
&h(\und{a}) \in I(a_0), \text{ for } \und{a} = (a_0,a_1,\dots) \in \Sigma^+\\
&h\circ\sigma = g \circ h.
\endalign
$$

Let $r$ be a real number $> 1$, or $r=+\infty$. The space of $C^r$ expansive maps of type $\Sigma$, endowed with the $C^r$ topology, will be denoted by $\Om_\Sigma^r$\,. The union $\Om_\Sigma = \dsize\bigcup_{r>1} \Om_\Sigma^r$ is endowed with the inductive limit topology.


\vglue .2in

\noindent{\bf 1.2}\quad For $(a,b) \in \B$, let
$$
f_{a,b} = \big[g\big|_{I(a,b)}\big]^{-1}\,;
$$
this is a contracting diffeomorphism from $I(b)$ onto $I(a,b)$. If $\und{a} =
(a_0,\dots,a_n)$ is a word of $\Sigma$, we put
$$
f_{\und{a}} = f_{a_0,a_1} \circ\dots\circ f_{a_{n-1}a_n}\,;
$$
this is a diffeomorphism from $I(a_n)$ onto a subinterval of $I(a_0)$ that we
denote by $I(\und{a})$.

\vglue .2in

\noindent{\bf 1.3}\quad Let $\Sigma^- = \{(\te_n)\_{n\le0}\,, (\te_i,\te_{i+1})
\in \B \text{ for } i < 0\}$. We equip $\Sigma^-$ with the following
ultrametric distance: for $\und{\te} \ne \und{\widetilde\te} \in \Sigma^-$, set
$$
d(\und{\te},\und{\widetilde\te}) =
\cases
1 \quad\text{if}\quad \te_0 \ne \widetilde\te_0\\
|I(\und{\te} \wedge \und{\widetilde\te})|\quad\text{otherwise}
\endcases
$$
where $\und{\te} \wedge \und{\widetilde\te} = (\te_{-n},\dots,\te_0)$ if
$\widetilde\te_{-j} = \te_{-j}$ for $0 \le j \le n$ and $\widetilde\te_{-n-1}
\ne \te_{-n-1}$\,.

Now, let $\und{\te} \in \Sigma^-$; for $n > 0$, let $\und{\te}^n = (\te_{-n},\dots,\te_0)$, and let $B(\und{\te}^n)$ be the affine map from
$I(\und{\te}^n)$ onto $I(\te_0)$ such that the diffeomorphism $k_n^{\und{\te}}
= B(\und{\te}^n) \circ f_{\und{\te}^n}$ is orientation preserving.

We have the following well-known result (see [Su]):

\vglue .2in

\noindent{\bf Proposition}. Let $r \in (1,+\infty)$, $g \in \Om_\Sigma^r$\,.

\item{\bf 1.} For any $\und{\te} \in \Sigma^-$, there is a diffeomorphism $k^{\und{\te}} \in \Diff_+^r(I(\te_0))$ such that $k_n^{\und{\te}}$ converge to $k^{\und{\te}}$ in $\Diff_+^{r'}(I(\te_0))$, for any $r'< r$, uniformly in $\und{\te}$. The convergence is also uniform in a neighbourhood of $g$ in $\Om_\Sigma^r$\,.

\item{\bf 2.} If $r$ is an integer, or $r = +\infty$,\,\,\,$k_n^{\und{\te}}$ converge to $k^{\und{\te}}$ in $\Diff_+^r(I(\te_0))$. More precisely, for every $0 \le j \le r-1$, there is a constant $C_j$ (independent on $\und{\te}$ such that
$$
|D^j\,\Log\,D\big[k_n^{\und{\te}} \circ (k^{\und{\te}})^{-1}\big](x)| \le C_j|I(\und{\te}^n)|.
$$
It follows that $\und{\te} \to k^{\und{\te}}$ is Lipschitz in the following sense: for $\te_0 = \widetilde\te_0$\,, we have
$$
|D^j\,\Log\,D\big[k^{\und{\widetilde\te}} \circ (k^{\und{\te}})^{-1}\big](x)| \le C_j\,d(\und{\te}, \und{\widetilde\te}).
$$

\vglue .2in

\noindent{\bf 1.4}\quad For $\und{\te} \in \Sigma^-$ and $\und{a}$ a word in $\Sigma$ starting with $a_0 = \te_0$\,, we denote:
$$
\align
&K(\und{a}) = K \cap I(\und{a}),\\
&I^{\und{\te}}(\und{a}) = k^{\und{\te}}(I(\und{a})),\\
&K^{\und{\te}}(\und{a}) = k^{\und{\te}}(K(\und{a})).
\endalign
$$
The different $K^{\und{\te}}$ are related in the following way: let
$F^{\und{\te}}$ be the affine map from $I(\te_0)$ onto $I^{\sigma^{-
1}(\und{\te})}(\te_{-1},\te_0)$ with the same orientation than 
$f_{\te_{-1}\te_0}$\,. Then, on $I(\te_0)$:
$$
F^{\und{\te}} \circ k^{\und{\te}} = k^{\sigma^{-1}(\und{\te})} \circ f_{\te_{-1}\te_0}\,.
$$
If we define
$$
\align
&\I = \{(\und{\te},x), \und{\te} \in \Sigma^-, x \in I(\te_0)\}\\
&\K = \{(\und{\te},x), \und{\te} \in \Sigma^-, x \in K^{\und{\te}}(\te_0)\}\\
&F(\und{\te},x) = (\sigma^{-1}(\und{\te}), F^{\und{\te}}(x)),
\endalign
$$
we have a fiber-contracting map $F$ such that
$$
\K = \bigcap_{n\ge0} F^n(\I).
$$
Moreover, if we define $H\colon \Sigma \to \K$ by
$$
H(\und{\te}) = (\pi^-(\und{\te}),\,\, k^{\pi^-(\und{\te})} \circ
h(\pi^+(\und{\te})))
$$
where $\pi^-$, $\pi^+$ are the natural projections of $\Sigma$ onto $\Sigma^-$,
$\Sigma^+$ respectively, the map $H$ is an homeomorphism which satisfies
$$
H \circ \sigma^{-1} = F \circ H.
$$

\vglue .2in

\noindent{\bf 1.5}\quad Let $r \in (1,+\infty]$. For $a \in \AA$, denote by $\P^r(a)$ the space of $C^r$-embeddings of $I(a)$ into $\re$, endowed with the $C^r$ topology. The affine group $\Aff(\re)$ acts by composition on the left on $\P^r(a)$, the quotient space being denoted by $\ov\P^r(a)$. We also consider $\P(a) = \dsize\bigcup_{r>1} \P^r(a)$ and $\ov\P(a) = \dsize\bigcup_{r>1} \ov\P^r(a)$, endowed with the inductive limit topologies.

Let $(a_0,a_1) \in \B$, and $g \in \Om_\Sigma^r$\,. We define the renormalization operator
$$
\align
T_{a_1a_0}^g\colon \P^r(a_0) &\to \P^r(a_1)\\
h &\mapsto h\circ f_{a_0,a_1}\,.
\endalign
$$
If $\und{a} = (a_0,\dots,a_n)$ is a word in $\Sigma$, we write $T_{\und{a}}^g\colon \P^r(a_0) \to \P^r(a_n)$ for $T_{a_na_{n-1}}^g \circ\dots\circ T_{a_1,a_0}^g$\,. We have
$$
T_{\und{a}}^g(h) = h\circ f_{\und{a}}\,.
$$
The renormalization operators commute to the action of the affine group, hence they are defined on the quotient spaces $\ov\P^r(a)$ as well.

We can reinterpret the proposition above as follows. For $\und{\te} \in \Sigma^-$, the sequence $T_{\und{\te}^n}^g(h_n)$\quad $(h_n \in \P^r(\te_{-n}))$ converge (in the $\C^{r'}$ topology, $r'< r$) to a limit in $\ov\P^r(\te_0)$, provided the $h_n$ stay bounded. The limit is independent of the $h_n$ and the convergence is uniform in $\und{\te}$ and bounded subsets of the $\P^r(a)$.

On the limit set of the renormalization operators, we can also view the renormalization dynamics as follows. Let $\A = \{(\und{\te},A)\}$, where $\und{\te} \in \Sigma^-$ and $A$ is now an {\it affine\/} embedding of $I(\te_0)$ into $\re$. We have a canonical map
$$
\align
\A &\to \P^r = \bigcup_{\AA} \P^r(a)\\
(\und{\te},A) &\mapsto A\circ k^{\und{\te}}\quad (\in \P^r(\te_0)).
\endalign
$$
We can lift the action of the renormalization operators to $\A$: if $(\te_0,\te_1) \in \B$, we set
$$
T_{\te_1,\te_0}(\und{\te},A) = (\und{\te}\te_1, A \circ F^{\und{\te}\te_1}).
$$
At this level, the various $T_{\te_1,\te_0}$ are just the inverse branches of the map $S$:
$$
S(\und{\te},A) = (\sigma^{-1}(\und{\te}), A \circ (F^{\und{\te}})^{-1}).
$$
The affine group $\Aff(\re)$ still acts by composition on the left on $\A$ (fiberwise onto $\Sigma^-$).

\vglue .5in

\noindent{\bf \S2.\quad Intersections of regular Cantor sets: setting and statement of the main result}.

\vglue .2in

\noindent{\bf 2.1}\quad In this section, we assume we are given two sets of data $(\AA,\B,\Sigma,g)$, $(\AA',\B',\Sigma',g')$ defining regular Cantor sets $K$, $K'$.

We define as in \S1.5 the spaces $\P = \dsize\bigcup_{\AA} \P(a)$ and $\P'= \dsize\bigcup_{\AA'} \P(a')$.

A pair $(h,h')$, $(h \in \P(a), h'\in \P '(a'))$ is called a {\it smooth configuration\/} for $K(a)$, $K'(a')$. Actually, rather than working in the product $\P \times \P'$, it is better to go to the quotient $Q$ by the diagonal action of the affine group $\Aff(\re)$. Elements of $Q$ are called {\it smooth relative configurations\/} for $K(a)$, $K'(a')$.

We say that a smooth configuration $(h,h') \in \P(a)\times \P(a')$ is
\item{--} {\it linked\/} if $h(I(a)) \cap h'(I(a')) \ne \emptyset$;
\item{--} {\it intersecting\/} if $h(K(\und{a})) \cap h'(K(\und{a}')) \ne \emptyset$;
\item{--} {\it stably intersecting\/} if it is still intersecting when we perturb it in $\P\times\P'$, and we perturb $(g,g')$ in $\Om_\Sigma \times \Om_{\Sigma'}$\,.

\noindent All these definitions are invariant under the action of the affine group, and therefore make sense for smooth relative configurations.

The following remark is obvious, but underlines the fundamental role played by the renormalization operators.

A smooth relative configuration $(h_0,h_0')$ is intersecting if and only if there exists a sequence $(h_n, h_n')$, {\it relatively compact in\/} $Q$, such that $(h_{n+1}, h_{n+1}')$ is obtained from $(h_n, h_n')$ by applying one of the renormalization operators (associated to $g$ or
$g'$). In particular,this condition implies that the configuration
$(h_0,h_0')$ is linked.

\vglue .2in

\noindent{\bf 2.2}\quad As in 1.5, we can introduce the spaces $\A$, $\A'$ associated to the limit geometries of $g$,\,\,$g'$ respectively. We denote by $\C$ the quotient of $\A\times\A'$ by the diagonal action on the left of the affine group. An element of $\C$, represented by $(\und\te,A) \in \A$,\,\, $(\und{\te}', A') \in \A'$, is called a relative configuration of the limit geometries determined by $\und{\te}$, $\und{\te}'$. We have canonical maps
$$
\align
\A\times\A'&\to \P\times\P'\\
\C &\to Q
\endalign
$$
which allow to define linked, intersecting, and stably intersecting configurations at the level of $\A\times\A'$ or $\C$.

\vglue .2in

\noindent{\bf Remark}: For a configuration $((\und{\te}, A), (\und{\te}',A'))$ of limit geometries, one could also consider the {\it weaker\/} notion of stable intersection, obtained by considering perturbations of $g$, $g'$ in $\Om_\Sigma \times \Om_{\Sigma'}$ and perturbations of $(\und{\te},A)$, $(\und{\te}',A')$ in $\A\times\A'$. We do not know of any example of expansive maps $g$, $g'$, and configurations $(\und{\te},A)$, $(\und{\te}',A')$ which are stably intersecting in the weaker sense but not in the stronger sense.

\vglue .2in

\noindent{\bf 2.3}\quad When looking for stably intersecting configurations, the following notion is crucial.

A compact set $\L$ in $\C$ is {\it recurrent\/} if for every $u \in \L$ we can find $v \in \tni \L$ and $\ell,\ell'\ge 0$ with $\ell+\ell'> 0$ and $u = S^\ell S^{\prime{\ell'}}(v)$. In other terms, if $(\und{\te},A)$, $(\und{\te}',A')$ represents $u$, we can find words $\und{a} = (a_0,\dots,a_\ell)$, $\und{a}'= (a_0',\dots,a_{\ell'}')$ in 
$\Sigma$, $\Sigma'$ respectively, with $a_0=\te_0$\,, $a_0'= \te_0'$\,, such that $(T_{\und{a}}(\und{\te},A), T_{\und{a}'}'(\und{\te}',A')) = v$ belongs to $\tni \L$.

We say that $\L$ is {\it immediately recurrent\/} if moreover we can take $\ell+\ell'= 1$ (for any $u \in \L$).

Let $\L$ be a recurrent compact set. Then one can find finitely many compact sets $\L_1,\dots,\L_N$\,, and words $\und{a}^i$, $\und{a}^{\prime i}$, $1 \le i \le N$ (with, for every $i$, at least one having more than one letter) such that

\item{(i)}\quad $\dsize\bigcup_{1\le i\le N} \L_i$ is a neighbourhood of $\L$;
\item{(ii)}\quad $T_{\und{a}^i}$\,\,$T'_{\und{a}^{\prime i}}$ is defined on $\L_i$ and sends $\L_i$ into $\tni \L$.

\noindent From this, we deduce immediately the following properties of recurrent compact sets.

\item{1.} Any recurrent compact set is contained in an immediately recurrent compact set.
\item{2.} Any recurrent (resp. immediately recurrent) compact set for $g$, $g'$ is still recurrent (resp. immediately recurrent) for $\tilde g, \tilde g'$ in a neighbourhood of $(g,g')$ in $\Om_\Sigma \times \Om_{\Sigma'}$\,.
\item{3.} Any relative configuration contained in a recurrent compact set is intersecting.

\vglue .1in

\noindent The last property can be strengthened:

\vglue .2in

\noindent{\bf Proposition}: Any relative configuration (of limit geometries) contained in a recurrent compact set is stably intersecting.

\vglue .1in

\noindent{\bf Proof}: By property 1 above, we may assume that the given configuration $u = [(\und{\te},A), (\und{\te}',A')]$ is contained in an immediately recurrent compact set $\L$. We first show that any small perturbation of $(A\circ k^{\und{\te}}, A'\circ k^{\prime\und{\te}'})$ in $\P(\te_0) \times \P'(\te_0')$ is still an intersecting smooth configuration for $(g, g')$. Such a perturbation can be written $(h\circ A \circ k^{\und{\te}}, h' \circ A'\circ k^{\prime\und{\te}'})$, with $h$, $h'$ close to the identity in some $C^r$-topology, $r > 1$. Applying one of the renormalization operators, and dividing $h$ into a translation and a translation-free part, we obtain {\it in\/} $Q$\quad $(\tilde h\circ \widetilde A \circ k^{\und{\widetilde\te}}, \tilde h'\circ \widetilde A'\circ k^{\prime\und{\widetilde\te}'})$, with $\tilde h$, $\tilde h'$ closer to the identity than were $h$, $h$', and $[(\widetilde A, \widetilde\te), (\widetilde A', \widetilde\te')]$ close (in $\C$) to the image of $u$ under the same renormalization operator.

It follows, by the last remark in 2.1, that this smooth configuration is intersecting.

Finally, the same argument works uniformly in a neighbourhood of $(g, g')$ in $\Om_\Sigma \times \Om_{\Sigma'}$ (see property 2 above); this concludes the proof. \qquad\quad \qed

\vglue .2in

\noindent{\bf 2.4}\quad Our main result is now the following.

\vglue .2in

\noindent{\bf THEOREM}: Let $g$, $g'$ be two $C^\infty$ expansive maps as above. Assume that the associated Cantor sets satisfy
$$
HD(K) + HD(K') \ge 1.
$$

Then, we can find, arbitrarily close to $g$, $g'$ (in the $C^\infty$-topology), perturbations $\tilde g$, $\tilde g'$ having a {\it non empty recurrent compact set\/} of relative configurations.

\vglue .1in

\noindent{\bf Remark}: In the case where both $g$ and $g'$ are orientation-preserving (equivalently, all periodic orbits of $g$ and $g'$ have positive eigenvalues), one can moreover find such a recurrent compact set which meets the two possible relative orientations (of $A$, $A'$); this will be apparent from the proof.

We consider the following subset $V$ of $\Om_\Sigma \times \Om_{\Sigma'}$\,. A pair $(g, g')$ belongs to $V$ if for any $[(\und{\te},A), (\und{\te}',A')] \in \A \times \A'$ there is a translation $R_t$ (in $\re$) such that $(R_t\circ A \circ k^{\und{\te}}, A'\circ k^{\prime\und{\te}'})$ is a stably intersecting configuration.

\vglue .2in

\noindent{\bf THEOREM}:

1.\quad $V$ is open in $\Om_\Sigma\times\Om_{\Sigma'}$\,, and $V \cap (\Om_\Sigma^\infty \times \Om_{\Sigma'}^\infty)$ is dense (for the $C^\infty$-topology) in the set $\{(g,g'), HD(K) + HD(K') > 1\}$.

2.\quad Let $(g,g') \in V$. There exists $d^* < 1$ such that for any $(h,h') \in \P\times\P'$, the set 
$$
\I_s = \{t\in\re, (R_t\circ h, h') \text{ is a stably intersecting smooth configuration for } (g,g')\}
$$
is (open and) dense in
$$
\I = \{t\in\re, (R_t\circ h, h') \text{ is an intersecting smooth configuration for } (g,g')\}
$$
and moreover $HD(\I-\I_s) \le d^*$. The same $d^*$ is also valid for $(\tilde g, \tilde g')$ in a neigbourhood of $(g,g')$ in $\Om_{\Sigma} \times \Om_{\Sigma'}$\,.

\vglue .1in

\noindent{\bf Proof}: Let us first prove that $V$ is open. For $\la \in \re^*$, denote by $H_\la$ the homothety $x \mapsto \la x$. Let $(g,g') \in \Om_\Sigma \times \Om_{\Sigma'}$\,, and $R > 1$ be larger than the supremum of the derivatives of the expansive maps $g$, $g'$. Then $(g, g')$ belongs to $V$ if and only if for every $\und{\te} \in \Sigma^-$,\,\,\, $\und{\te}'\in \Sigma^{\prime -}$,\,\,\, $\la \in J_R = [-R, -R^{-1}] \cup [R^{-1}, R]$, there exists $t \in \re$ such that $(R_t\circ k^{\und{\te}}, H_\la \circ k^{\und{\te}'})$ is a stably intersecting configuration. Using the compactness of the set of $(\und{\te}, \und{\te}',\la)$ under consideration, we see not only that $V$ is open but for $(g,g') \in V$ the following is true: there exist $\delta > 0$, a neighbourhood $W$ of $(g,g')$ in $\Om_\Sigma\times\Om_{\Sigma'}$ and a neighbourhood $Z$ of the identity in $\dsize\bigcup_{r > 1} \Diff^r(\re)$ such that for any $(\und{\te}, \und{\te}',\la) \in \Sigma^- \times \Sigma^{\prime -}\times J_R$ there is an interval $J$ of length $\delta$ with $(R_t\circ\tilde h\circ k^{\und{\te}}, \tilde h'\circ k^{\und{\te}'})$ being a stably intersecting configuration for all $t \in J$,\,\, $\tilde h, \tilde h'\in Z$ and $(\tilde g, \tilde g') \in W$.

Assertion 2 in the theorem is now an easy consequence of the last statement. Let $(g,g') \in V$, $(h,h') \in \P\times\P'$, $t \in \I$. Let $\ve > 0$ be small; let $\und{a}$, $\und{a}'$ be words in $\Sigma$, $\Sigma'$ respectively such that $I(\und{a})$, $I(\und{a}')$ have size of order $\ve$, and such that $h(K(\und{a})) \cap h'(K(\und{a}')) \ne \emptyset$. We apply (in the quotient space $Q$) $T_{\und{a}} T_{\und{a}'}'$ to $(R_t\circ h, h')$ and we get an intersecting configuration of the form $(\tilde h \circ k^{\und{\te}}, \tilde h'\circ H_\la \circ k^{\und{\te}'})$, where $\und{\te}$,  $\und{\te}'$ end with $\und{a}$, $\und{a}'$ respectively and $\tilde h, \tilde h'\in Z$ if $\ve$ is small enough. The statement above now shows that there is an interval of length $c\delta\ve$ (for some fixed $c > 0$) contained in $\I_s \cap [t-\ve, t+\ve]$. But this last property at the same time guarantees that $\I_s$ is dense in $\I$, and that $HD(\I-\I_s)$ is at most $d^* < 1$, where $d^*$ depends only on $c\delta$ (cf. [PT2], p.79, prop. 6).

To show denseness of $V$, we will show that if $(g,g')$ admits a non empty recurrent compact set $\L$ (meeting both orientations if $g$ and $g'$ are orientation-preserving, see remark after theorem 2.4), and if there are periodic points $x$, $x'$ of period $n$, $n'$ of $g$, $g'$ respectively such that
$$
\frac{\Log |Dg^n(x)|}{\Log |Dg^{\prime n'}(x')|}
$$
is irrational, then $(g,g') \in V$. Indeed, let $\und{a}$ (resp. $\und{a}')$ the word in $\Sigma$ (resp. $\Sigma')$ of length $n+1$ (resp. $n'+1$) such that $f_{\und{a}}(x) = x$ (resp. $f_{\und{a}'}'(x') = x')$. Let $N$ be the set of $(\und{\te}, \und{\te}',\la) \in \Sigma^- \times \Sigma^{\prime -} \times \re^*$ such that there exists $t \in \re$ for which $(R_t \circ k^{\und{\te}}, H_\la \circ k^{\und{\te}'})$ have stable intersection.

From the existence of the compact recurrent set $\L$, we deduce that $N$ is non empty, and even that for any given $(\und{\te}, \und{\te'}) \in \Sigma^- \times \Sigma^{\prime -}$ there are $\la_1 < 0 < \la_2$ such that $(\und{\te}, \und{\te'}, \la_i) \in N$ for $i=1,2$. But then, setting $\und{\bar a} = (\dots\und{a}, \und{a}, \und{a})$ and $\und{\bar a}' = (\dots \und{a}', \und{a}', \und{a}')$, the hypothesis on the eigenvalues imply that $(\und{\bar a}, \und{\bar a}',\la) \in N$ for all $\la \in \re^*$. Using again the renormalization operators, we finally get that $N = \Sigma^- \times \Sigma^{\prime -} \times \re^*$. The proof is complete. \qquad\quad\qed 

\vglue .1in

\noindent{\bf 2.5}\quad Let $\tilde d > 0$.

A smooth configuration $(h,h') \in \P(a)\times\P(a')$ has 
$\tilde d$-stable intersection if the Hausdorff dimension of $h(K(a)) \cap h'(K(a'))$ is $> \tilde d$, and this property still holds when we perturb $(h,h') \in \P(a)\times\P(a')$ and $(g,g')$ in $\Om_\Sigma \times \Om_{\Sigma'}$\,.

In relation to this property, we say that a compact subset $\L \subset \C$ is $\tilde d$-recurrent if the following holds: for any $((\und{\te},A), (\und{\te}',A'))$ representing a point $u \in \L$, we can find pairs $(\und{a}^1, \und{a}^{'1}),\dots,(\und{a}^k, \und{a}^{' k})$ for words such that

\item{i)} the first letter of $\und{a}î$ (resp. $\und{a}^{'i}$) is $\te_0$\,;
\item{ii)} the intervals $I(\und{a}^i)$ are disjoint;
\item{iii)} $(T_{\und{a}^i}(\und{\te},A), T_{\und{a}^{'i}}'(\und{\te}',A'))$ represents a point in $\tni \L$ for $1 \le i \le k$;
\item{iv)} $\sum\limits_1^k |I^{\und{\te}}(\und{a}^i)|^{\tilde d} > 1$.

By the compactness of $\L$, we can find the $\und{a}^i$, $\und{a}^{'i}$ with uniformly bounded length,

\item{iv')} $\sum\limits_1^k |I^{\und{\te}}(\und{a}^i)|^{\tilde{d}'} \ge c > 1$,

\noindent with $\tilde{d}'$ slightly larger than $\tilde d$.

Then (iv) will still hold when we perturb $(g, g')$. On the other hand, from (iv'), we fairly easily deduce the

\vglue .2in

\noindent{\bf Proposition}: If a relative affine configuration belongs to a $\tilde d$-recurrent non empty compact subset of $\C$, it is a $\tilde d$-stable intersecting configuration.

On the other hand, we will indicate in the course of the proof of the first theorem in 2.4 that slight modifications give the following stronger result.


\vglue .2in

\noindent{\bf THEOREM}: Let $0 < \tilde d < HD(K) + HD(K')-1$. Then we can find arbitrarily small $C^\infty$-perturbations of $(g, g')$ which exhibit a non empty $\tilde d$-recurrent compact subset.

\vglue .3in

\noindent{\bf \S3.\quad Outline of the proof of the main theorem}.

\vglue .2in

\noindent{\bf 3.1}\quad Let $g$, $g'$ be expansive maps as above. If
$((\und{\te},A), (\und{\te}',A'))$ is a relative configuration, the affine map $A^{-1}\circ A'\colon \re\times\{\te_0'\} \to \re\times\{\te_0\}$ is well defined; the coefficient of $x$ is a number in $\re^*$ which is called the {\it relative scale\/} of the given relative configuration. We obtain thus the space
$\si = \Sigma^- \times \Sigma^{\prime -} \times \re^*$ of relative scales, which is canonically a quotient of the space $\C$ of relative configurations (under the actions of translations on $\A$, $\A'$, independently).

The renormalization operators $S$, $S'$ and their inverse branches
$T_{a_0,a_1}$\,, $T'_{a_0',a_1'}$ still act on $\si$.

Actually, let $\und{\te} \in \Sigma^-$, $\und{\te}' \in \Sigma^{\prime -}$,
and $\und{a}$ be a word in $\Sigma$ starting with $\te_0$\,. Let
$$
T_{\und{a}} = T_{a_n,a_{n-1}}\circ\dots\circ T_{a_1,a_0}\,.
$$
Then we have, for $s \in \re^*$
$$
T_{\und{a}}(\und{\te}, \und{\te}',s) = (\und{\te}a_1\dots a_n, \und{\te}', \ve
s |I^{\und{\te}}(\und{a})|^{-1})
$$
where $\ve = +1$ (resp. $-1$) if $f_{\und{a}}$ is orientation preserving (resp.
reversing).

Similarly, if $\und{a}'$ is a word in $\Sigma$ starting with $\te_0'$\,, we
have
$$
T'_{\und{a}'}(\und{\te},\und{\te}',s) = (\und{\te}, \und{\te}' a_1'\dots a_n',
\ve's |I^{\und{\te}'}(\und{a}')|).
$$

\vglue .2in

\noindent{\bf 3.2}\quad The first main ingredient in the proof of the main
theorem is a recurrence result for the action of renormalization on the space
$\si$ of relative scales; we now present this result, which we call the
Scale recurrence Lemma. Its proof is defferred to section 6.

We will always restrict our attention to a compact subset
$$
\si_R = \Sigma^- \times \Sigma^{\prime -} \times ([-R,-R^{-1}] \cup [R^{-1},
R])
$$
of $\si$, with $R$ conveniently large.

Fix once and for all a conveniently large constant $c_0$\,. The {\it size\/} of
a word $\und{a}$ in $\Sigma$ is the length $|I(\und{a})|$; we say that
$\und{a}$ has approximate size $\ro$ if
$$
c_0^{-1}\,\ro \le |I(\und{a})| \le c_0\ro
$$
and denote by $\Sigma(\ro)$ the set of those words. Its cardinality is of order
$\ro^{-d}$, where we set
$$
d = HD(K), \quad d' = HD(K').
$$
We define similarly $\Sigma'(\ro)$, with cardinality of order $\ro^{-d'}$. To
prove the Scale recurrence Lemma below, we need to assume that at least one of the two
regular Cantor sets $K$, $K'$ is non linear. More precisely, we assume that
there exist $\und{\te}^0,\und{\te}^1 \in \Sigma^-$, with $\te_0^0 = \te_0^1$\,,
and $x_0 \in K^{\und{\te}^0}(\te_0^0)$ such that
$$
|D\,\Log\,D(k^{\und{\te}^1} \circ (k^{\und{\te}^0})^{-1})(x_0)| \ne 0.
$$
Then, denoting $J_R = [-R,-R^{-1}] \cup [R^{-1},R]$, we have:

\vglue .2in

\noindent{\bf SCALE RECURRENCE LEMMA}. If $R$, $c_0$ are conveniently large, we can
find $c_1,c_2,c_3 > 0$ with the following properties: given $0 < \ro < 1$, and
a family $E(\und{a}, \und{a}')$ of subsets of $J_R$\,,\,\,\ $(\und{a},\und{a}')
\in \Sigma(\ro) \times \Sigma'(\ro)$, such that
$$
m(J_R-E(\und{a},\und{a}')) \le c_1\,, \qquad \forall\,(\und{a},\und{a}')
$$
we can find another family $E^*(\und{a},\und{a}')$ of subsets of $J_R$
satisfying:

\item{(i)} for any $(\und{a},\und{a}')$, $E^*(\und{a},\und{a}')$ is contained
in a $c_2\ro$-neighbourhood of $E(\und{a},\und{a}')$;
\item{(ii)} let $(\und{a},\und{a}') \in \Sigma(\ro) \times \Sigma'(\ro)$, $s
\in E^*(\und{a},\und{a}')$; we can find at least $c_3\,\ro^{-(d+d')}$ pairs
$(\und{b},\und{b}') \in \Sigma(\ro) \times \Sigma'(\ro)$ (with $\und{b}$,
$\und{b}'$ starting respectively with the last letter of $\und{a}$, $\und{a}'$)
such that, if $\und{\te} \in \Sigma^-$, $\und{\te}' \in \Sigma^{\prime -}$ end
respectively with $\und{a}$, $\und{a}'$ and
$$
T_{\und{b}} T_{\und{b}'}'(\und{\te}, \und{\te}',s) = (\und{\widetilde\te},
\und{\widetilde\te}',\tilde s)
$$
the $\ro$-neighbourhood of $\tilde s$ is contained in $E^*(\und{b}, \und{b}')$.

\vglue .2in

Observe that $\und{\widetilde\te}$, $\und{\widetilde\te}'$ above end
respectively with $\und{b}$, $\und{b}'$. 

Let $\si^*(\und{a},\und{a}')$ be the set of $(\und{\te},\und{\te}',s) \in
\si_R$ such that $\und{\te}$ ends with $\und{a}$, $\und{\te}'$ with $\und{a}'$
and $s \in E^*(\und{a}, \und{a}')$; set $\si^* =
\underset{\und{a},\und{a}'}\to{\bigcup}\,\si^*(\und{a}, \und{a}')$. What
property (ii) says is that there is a positive proportion of compatible
$(\und{b}, \und{b}') \in \Sigma(\ro) \times \Sigma'(\ro)$ such that the image
by the inverse branch $T_{\und{b}}T_{\und{b}'}$ is ``well inside'' $\si^*$.

\vglue .2in

\noindent{\bf 3.3}\quad The fibers of the quotient map: $\C \to \si$ are 
one-dimensional and have a canonical affine structure; moreover this bundle
map is trivializable; we choose now an explicit trivialization $\C \cong \si
\times \re$ in order to have a coordinate in each fiber.

For each $a \in \AA$ (resp. $a' \in \AA'$), we choose $\om(a) \in \Sigma^+$
starting with $a$ (resp. $\om(a') \in \Sigma^{\prime +}$ starting with $a'$).
Given a configuration represented by $(\und{\te},A)$, $(\und{\te}',A')$, we
normalize (by the action of the affine group) in order to obtain that $A'\colon
\re\times\{\te_0\} \to \re$ becomes a translation sending
$k^{\prime\und{\te}'}(h'(\om(\te_0')))$ to 0.

Then our coordinate $t((\und{\te},A), (\und{\te}',A'))$ on the fiber will be by
definition
$$
t = A(k^{\und{\te}}(h(\om(\te_0)))).
$$

With the chosen normalization, $A'(I(\te_0'))$ has size 1 while $A(I(\te_0))$
has size $|s|$.

\vglue .2in

\noindent{\bf 3.4}\quad The first step in the proof is to get sure that the
hypothesis of the Scale recurrence Lemma is satisfied (by either $g$ or $g'$), and
that $d+d' > 1$.

So we make if necessary a small perturbation and will now assume that these
hypotheses are satisfied by $g$, $g'$.

We want to perturb $g$, $g'$ (actually it will be sufficient to perturb $g$) in
the $C^\infty$-topology in order to create a non empty recurrent compact set of
relative configurations.

But a neighbourhood in the $C^\infty$ topology is a neighbourhood in some $C^k$
topology for finite $k$. We now fix such an integer $k \ge 2$.

The construction of the perturbation will depend on a scale parameter $\ro$;
the perturbation will become close to $g$ when $\ro$ is small.

In the following, we denoted such a scale by $\ro$, and it will be assumed to
be small.

\vglue .2in

\noindent{\bf 3.5}\quad The required perturbation for $g$ will be picked by a probabilistic argument out of a family of random perturbations that we will now construct.

We first pick a subset $\Sigma_0$ of $\Sigma(\ro^{1/k})$ such that
$$
K = \bigcup_{\und{a}\in\Sigma_0} K(\und{a})
$$
is a {\it partition\/} of $K$ into disjoint cylinders.

We then define $\Sigma_1$ as the subset of $\Sigma_0$ formed of the words
$\und{a} \in \Sigma_0$ such that no word in $\Sigma(\ro^{1/3k})$ appears twice
in $\und{a}$.

Let $c_4 > 1$ be a constant sufficiently close to 1 to have the following: let
$\widehat I(\und{a})$, for $\und{a} \in \Sigma_0$\,, be the interval with same
center as $I(\und{a})$ and $c_4^2$ times the size of $I(a)$; then the $\widehat
I(\und{a})$, $\und{a} \in \Sigma_0$\,, are pairwise disjoint.

We then choose, once and for all, a smooth even function $\chi\colon \re\to\re$
satisfying
$$
\align
&\chi(x) = 1 \quad\text{for}\quad |x| \le c_4\,,\\
&\chi(x) = 0 \quad\text{for}\quad |x| \ge c_4^2\,.
\endalign
$$

For $\und{a} \in \Sigma_1$\,, we define the vectorfield $X_{\und{a}}$\,, with support $\subset\, \widehat I(\und{a})$ by
$$
X_{\und{a}}(x) = c_5\,\ro^{(1+1/2k)}\,\chi(B(x)) \frac{\po}{\po x}
$$
where $c_5 > 0$ is a conveniently large constant, to be chosen later, and $B$
is an affine map sending $I(\und{a})$ onto $[-1,+1]$; there are two such maps,
but as $\chi$ is even, they give the same $X_{\und{a}}$\,.

The probability space underlying the famility of random perturbations is $\Om =
[-1,1]^{\Sigma_1}$, equipped with the normalized Lebesgue measure.

For $\und{\om} = (\om(\und{a}))\_{\und{a}\in\Sigma_1} \in \Om$, we define
$\Phi_{\und{\om}}$ to be the time-one map of the vector field $X_{\und{\om}} = 
-\sum\limits_{\und{a}} \om(\und{a})X_{\und{a}}$\,, and $g^{\und{\om}}$ to be $g
\circ \Phi_{\und{\om}}$\,. Our task will be to find $\und{\om} \in \Om$ such
that $(g^{\und{\om}},g')$ have a non empty compact recurrent set of relative
configurations.

\vglue .2in

\noindent{\bf 3.6}\quad We make now some easy remarks on the family
$g^{\und{\om}}$; the finer properties will be studied later.

\vglue .1in

\noindent{\bf 3.6.1}\quad By construction, the vectorfields $X_{\und{a}}$\,, 
$\und{a} \in \Sigma_1$\,, have disjoint supports.

The intervals $I(\und{a})$, $\und{a} \in \Sigma_1$\,, have approximate size
$\ro^{1/k}$; therefore the affine map $B$ entering the definition of $X_{\und
a}$ has derivative of the order of $\ro^{-1/k}$. It follows that, when $\ro$ is
small, $X_{\und{a}}$ is small in the $C^k$-topology. Then, the
$\Phi_{\und{\om}}$\,, for any $\und{\om} \in \Om$, are close to the identity in
the $C^k$-topology.

In particular, the estimates in proposition 1.3 for the $k^{\und{\te}}$ are
valid for $g^{\und{\om}}$ uniformly in $\und{\om}$, if we consider no more than
$k$ derivatives.

\vglue .1in

\noindent{\bf 3.6.2}\quad For $a \in \AA$, let
$$
\align
&x_{\min}(a) = \inf K(a),\\
&x_{\max}(a) = \sup K(a).
\endalign
$$
The finite set $\underset{a}\to{\bigcup}\,\{x_{\min}(a), x_{\max}(a)\}$ is
invariant under $g$. Let $x_0$ be a point in this set; if $\und{a}_0$ is the
element of $\Sigma_0$ such that $x_0 \in K(\und{a}_0)$,\,\, $\und{a}_0$ is the
initial part of an eventually periodic word in $\Sigma^+$. Therefore $\und{a}_0
\in \Sigma_0 - \Sigma_1$ if $\ro$ is small enough. It follows that
$g^{\und{\om}}$ coincides with $g$ in $I(\und{a}_0)$ (for any $\und{\om} \in
\Om$), and that the points $x_{\min}(a)$, $x_{\max}(a)$ are still the
``endpoints'' of the perturbed Cantor set $K_{\und{\om}}(a)$.

\vglue .1in

\noindent{\bf 3.6.3}\quad For $\und{a} \in \Sigma_0$\,, let $\widetilde
I(\und{a})$ be the $\ro$-neighbourhood of $I(\und{a})$.

Let $\und{a} \in \Sigma_0$\,, $a_{-1} \in \AA$ such that $(a_{-1},a_0) \in \B$;
let $\und{a}'$ be the initial part of $a_{-1}\und{a}$ that belongs to
$\Sigma_0$\,. We have $f_{a_{-1}a_0}(I(\und{a})) \subset I(\und{a}')$; because
the $f_{a,b}$ are contractions, there exists $c_6 > 0$ such that the 
$c_6\ro$-neighbourhood of $f_{a_{-1}a_0}(\widetilde I(\und{a}))$ is contained
in $\widetilde I(\und{a}')$.

On the other hand, for the perturbed inverse branch $f_{a_{-
1}a_0}^{\und{\om}}$\,, $\und{\om} \in \Om$, we will have, for any $x \in
\widetilde I(\und{a})$
$$
f_{a_{-1}a_0}(x) =
\cases
f_{a_{-1}a_0}(x) \quad\text{if}\quad \und{a}' \in \Sigma_0-\Sigma_1\\
f_{a_{-1}a_0}(x) + c_5\ro^{1+1/2k}\,\om(\und{a}') \text{ if } \und{a}' \in
\Sigma_1
\endcases
$$
(because $X_{\und{a}'}$ is constant on $\widetilde I(\und{a}'))$. This allows
to conclude that $f_{a_{-1}a_0}^{\und{\om}}(\widetilde I(\und{a})) \subset
\widetilde I(\und{a}')$ (if $\ro$ is small enough), and therefore that the
perturbed Cantor set $K_{\und{\om}}$ will be contained in
$\underset{\Sigma_0}\to{\bigcup}\,\widetilde I(\und{a})$.

\vglue .2in

\noindent{\bf 3.7}\quad We are now ready to give some information on the set
$\L = \L_{\und{\om}}$ of relative configurations for which we want to prove
recurrence for at least one $\und{\om} \in \Om$.

As indicated, this set will depend on $\und{\om}$, but only as far as the fiber
coordinate $t$ is concerned. The image of $\L_{\und{\om}}$ under the projection
map: $\C \to \si$ will be a subset $\widetilde\L$ of $\si$ independent of
$\und{\om}$.

Let $\Sigma_{nr}(\ro^3)$ be the subset of $\Sigma(\ro^3)$ formed by words
$\und{a}$ such that:

\item{(i)} no word $\und{b}\in\Sigma(\ro^{1/3k})$ appears twice in $\und{a}$;
\item{(ii)} if $\und{c} \in \Sigma(\ro^{1/6k})$ appears at the end of
$\und{a}$, then it does not appear elsewhere in $\und{a}$.

We next define $\Sigma_{nr}^-$ as the subset of $\Sigma^-$ formed by
$\und{\te}$ which end with a word in $\Sigma_{nr}(\ro^3)$. This is an open and
closed subset in $\Sigma^-$.

To define $\widetilde\L$, we will apply the Scale recurrence Lemma (with $\ro^{1/2}$
instead of $\ro$).

A family of subsets $E(\und{a}, \und{a}')$ of $J_R$\,, for $(\und{a}, \und{a}')
\in \Sigma(\ro^{1/2}) \times \Sigma'(\ro^{1/2})$ will be carefully constructed
in section 4, in relation to Marstrand's theorem, and it will satisfy the
hypothesis
$$
m(J_R - E(\und{a}, \und{a}')) \le c_1\,, \quad \forall(\und{a}, \und{a}').
$$
Then, the Scale recurrence Lemma gives us another family $E^*(\und{a}, \und{a}') \in
\Sigma(\ro^{1/2}) \times \Sigma'(\ro^{1/2})$, with the properties (i), (ii)
indicated in the statement of the lemma.

The set $\widetilde\L$ is defined to be the subset of $\si_R$ formed by the
$(\und{\te}, \und{\te}', s)$ such that $\und{\te} \in \Sigma_{nr}^-$\,, and
there exists $\und{a} \in \Sigma(\ro^{1/2})$, $\und{a}' \in \Sigma'(\ro^{1/2})$
with $s \in E^*(\und{a}, \und{a}')$ and $\und{\te}$, $\und{\te}'$ ending with
$\und{a}$, $\und{a}'$ respectively.

\vglue .2in

\noindent{\bf 3.8}\quad For every $(\und{\te}, \und{\te}', s)$ in
$\widetilde\L$, we will define in section 4, in relation to the
conclusions of Marstrand's theorem, a non empty subset
$L_{\und{\om}}^0(\und{\te}, \und{\te}',s)$, depending on $\und{\om} \in \Om$, of the fiber of $\C$ over $(\und{\te}, \und{\te}',s)$.

Let
$$
\L_{\und{\om}}^0 = \{(\und{\te}, \und{\te}', s,t), (\und{\te}, \und{\te}',s) \in \widetilde
\L, t \in L_{\und{\om}}^0(\und{\te}, \und{\te}',s)\};
$$
consider next the $\ro$-neighbourhood $\L_{\und{\om}}^1$ of $\L_{\und{\om}}^0$ in $\widetilde\L \times \re$:
$$
\align
&\quad\L_{\und{\om}}^1 = \{(\und{\te}, \und{\te}',s,t), (\und{\te}, \und{\te}',s) \in \widetilde\L \text{ and } \exists\, (\und{\te}_0, \und{\te}_0',s_0,t_0) \in \L_{\und{\om}}^0\\
&\text{with } d(\und{\te}, \und{\te}_0) < 2\ro^{5/2}, d(\und{\te}', \und{\te}_0') < 2\ro^{5/2}, |s-s_0| < \ro, |t-t_0| < \ro\}. 
\endalign
$$
Fix $u = (\und{\te}, \und{\te}',s,t) \in \widetilde\L\times\re$.
We define two subsets $\Om^0(u)$, $\Om^1(u)$ of $\Om$. First,
$$
\Om^1(u) = \{\und{\om} \in \Om, (\und{\te}, \und{\te}',s,t) \in
\L_{\und{\om}}^1\}.
$$
Second, $\Om^0(u)$ is the set of $\und{\om} \in \Om$ such that there exists $\und{b} \in \Sigma(\ro)$, $\und{b}' \in \Sigma'(\ro)$, with $b_0 = \te_0$\,, $b_0' = \te_0'$ and the image $T_{\und{b}} T_{\und{b}'}'(u) = (\und{\widetilde\te}, \und{\widetilde\te}',\tilde s, \tilde t$) satisfies:

\item{(i)} for any $\tilde s_1$ with $|\tilde s-\tilde s_1| < \dfrac 12 \ro^{1/2}$, we have $(\und{\widetilde\te}, \und{\widetilde\te}', \tilde s_1) \in \widetilde\L$;
\item{(ii)} $\tilde t \in L_{\und{\om}}^0(\und{\widetilde\te},
\und{\widetilde\te}',\tilde s)$.

[The renormalization operators $T_{\und{b}}$\,, $T_{\und{b}'}$ depend on $\und{\om}$; the dependence is not indicated in the notations to keep the text readable].

The following crucial estimate will be proven in section 5.

\vglue .2in

\noindent{\bf Proposition}: Assume that $c_5$ is chosen conveniently large.
Then there exists $c_7 > 0$, such that, for any $u \in \widetilde\L\times\re$, one has
$$
\bp(\Om^1(u) - \Om^0(u)) \le \exp\big(-c_7\ro^{-\frac{1}{2k}(d+d'-1)}\big).
$$

\comment
%ACRESCIMO
\vglue .2in

\noindent{\bf 3.8.1}\quad Fix $0 < \tilde d < d+d'-1$, and let $d^* = \dfrac 12 (\tilde d+d+d'-1)$. In section 4.5.1 we will define sets $\ov L_{\und{\om}}^0(\und{\te}, \und{\te}',s)$ on the fiber of $C$ over $(\und{\te}, \und{\te}',s)$. Let $\ov{\Cal L}_{\und{\om}}^0 = \{(\und{\te}, \und{\te}',s,t), (\und{\te}, \und{\te}',s) \in \widetilde L$;\linebreak $t \in \ov L_{\und{\om}}^0(\und{\te},\und{\te}',s)\}$ and $\ov{\Cal L}_{\und{\om}}^1$ the $\ro$-neighbourhood of $\ov{\Cal L}_{\und{\om}}^0$\,, as before. Fixed $U = (\und{\te}, \und{\te}',s,t) \in \ov{\Cal L} \times \re$ we define $\ov\Om^1(U) = \{\und{\om} \in \Om, (\und{\te}, \und{\te}',s,t) \in \ov{\Cal L}_{\und{\om}}^1\}$ and $\ov\Om^0(U)$ is the set of $\und{\om} \in \Om$ such that there exist $M = [\ro^{-d^*}]$ pairs $(\und{b}_i, \und{b}_i')$, $1 \le i \le N$ satisfying condition i) of the definition of $\Om^0(U)$ and the condition 

\item{ii')} $\tilde t_i \in \ov L_{\und{\om}}^0(\und{\widetilde\te}_i, \und{\widetilde\te}_i', \tilde s_i)$, \quad $1 \le i \le M$.

In section 4 we will see that the proposition 4.5 holds for the $\ov L_{\und{\om}}$ instead of the $L_{\und{\om}}$\,, and it implies that $\Bbb{P}(\ov\Om^1(U) - \ov\Om^0(U)) \le \ro^{-d^*}\, \exp(-\bar c_7 \ro^{-\frac{1}{2k}(d+d'-1)})$, as we shall see in the comments at the end of section 4.5.1.

This gives a $\tilde d$-recurrent compact set $\Cal L$ for most of the perturbations, and this proves Theorem 2.6.
%FIM DO ACRESCIMO
\endcomment

Basically, the reason for this inequality is the following: the sets $L(\und{\te}, \und{\te}',s)$ are formed of values of $t$ such that $T_{\und{b}}T_{\und{b}'}'(\und{\te}, \und{\te}',s,t)$ is bounded for many pairs $(\und{b},\und{b}') \in \Sigma(\ro)\times\Sigma'(\ro)$. It will follow from Marstrand's theorem that the Lebesgue measure of $L(\und{\te}, \und{\te}',s)$ is uniformly bounded from below. So we can expect for fixed $(\und{b}, \und{b}')$ the probability (in $\und{\om})$ that $T_{\und b} T_{\und{b}'}'(\und{\te}, \und{\te}',s,t)$ is right to be bounded from below. When $(\und{b},, \und{b}')$ vary, we will be able to find $c\,\ro^{-\frac{1}{2k}(d+d'-1)}$ such pairs for which we have rough independence, which `` explains'' the inequality in the proposition.

\vglue .2in

\noindent{\bf 3.9}\quad With this estimate, the end of the proof of the theorem is not difficult.

The sets $L_{\und{\om}}^0(\und{\te}, \und{\te}',s)$ will always satisfy
$$
t \in L_{\und{\om}}^0(\und{\te}, \und{\te}',s) \Rightarrow |t| \le 1+|s| \le 1+R.
$$
We choose finite subsets $\Delta_0$\,, $\Delta_0'$\,, $\Delta_2$ of
$\Sigma_{nr}^-$\,, $\Sigma^{\prime -}$, $[-(1+R+2\ro), (1+R+2\ro)]$
respectively such that:

\item{--} $\Delta_0$ is $\ro^{5/2}$ dense in $\Sigma_{nr}^-$\,, $\#\Delta_0 \le c_8\ro^{-\frac 52d}$;
\item{--} $\Delta_0'$ is $\ro^{5/2}$ dense in $\Sigma^{\prime -}$\,,
$\#\Delta_0' \le c_8\ro^{-\frac 52d'}$;
\item{--} $\Delta_2$  is $\ro^{5/2}$ dense in $[-(1+R+2\ro), 1+R+2\ro]$,
$\#\Delta_2 \le c_8\ro^{-\frac 52}$.

\noindent Then, for each $(\und{\te}, \und{\te}') \in \Delta_0 \times
\Delta_0'$\,, we choose a finite subset $\Delta_1(\und{\te}, \und{\te}')$ of
the fiber of $\widetilde\L$ over $(\und{\te}, \und{\te}')$, which is
$\ro^{5/2}$ dense in this fiber, and has cardinality $\le c_8\ro^{-5/2}$. Let then
$$
\Delta = \{u = (\und{\te}, \und{\te}', s, t), \und{\te} \in \Delta_0,
\und{\te}' \in \Delta_0', t \in \Delta_2, s \in \Delta_1(\und{\te},
\und{\te}')\}.
$$
One has
$$
\#\Delta \le c_8^4\, \ro^{-(5+\frac 52 (d+d'))} \le c_8^4\, \ro^{-10}.
$$
Now, if $\ro$ is small enough, we have
$$
c_8^4\,\ro^{-10}\,\exp\big(-c_7\, \ro^{-\frac{1}{2k}(d+d'-1)}\big) < 1,
$$
and therefore we can find $\und{\om}_0 \in \Om$ such that, for any $u \in \Delta$, we have either $\und{\om}_0 \notin \Om^1(u)$ or $\und{\om}_0 \in \Om^0(u)$ (or both).

\comment
\vglue .2in

\noindent{\bf 3.9.1}\quad If we have a differentiable family of Cantor sets $K_\mu$ defined by $g_\mu$\,, $\mu \in [-1,1]^n$, we can consider the family of perturbations of $(g_\mu)_{\mu\in[-1,1]^n}$\,, $\big((g_\mu^{\und{\om}})_{\mu\in[-1,1]^n}\big)_{\und{\om}\in\Om}$ defined by $g_\mu^{\und{\om}} = g_\mu \circ \phi_{\und{\om}}$\,. The estimate $c_8^4\,\ro^{-10-5n/2}\,\exp(-c_7\,\ro^{-\frac 12k(d+d'-1)}) \ll 1$ implies as before that an open and dense set of families $G\colon [-1,1]^n \to \{(K,K')$ regular Cantor sets $\mid HD(K) + HD(K') > 1\}$ have image contained in the open set $V$ of Proposition 2.4.1. The openess is automatic. The denseness follows from the above estimate, the differentiable dependence of the limit geometries in $\mu$ and from the fact that generically, for each $\mu$ there are periodic points of $K_\mu$ such that the ratio of the logarithms of the correspondent eigenvalues is irrational. The two first facts give a compact recurrent invariant set for each $\mu$ for most $\und{\om} \in \Om$, and the last one assures that the image is contained in $V$ provided we have compact invariant sets by renormalizations.
\endcomment

\vglue .2in

\noindent{\bf 3.10}\quad Fix $\und{\om}_0$ as above. Define $\L =
\L_{\und{\om}_0}$ as follows:
$$
\align
\L &= \{(\und{\te}, \und{\te}',s,t) \in \widetilde\L\times\re,\,\,\,
\exists\,(\und{\widetilde\te}, \und{\widetilde\te}', \tilde s, \tilde t) \in \widetilde\L \times \re\\
&\qquad \text{ with } d(\und{\te}, \und{\widetilde\te}) \le \ro^{5/2}, \,\,\,d(\und{\te}', \und{\widetilde\te}') \le \ro^{5/2},\\
&\qquad |s-\tilde s| \le \ro/2, \quad |t-\tilde t| \le \ro/2\\
&\qquad\quad \text{and } \tilde t \in  L_{\und{\om}_0}^0(\und{\widetilde\te},
\und{\widetilde\te}',\tilde s)\}.
\endalign
$$

We claim that for $(g^{\und{\om}_0},g')$, \,\,\, $\L$ is a non empty compact recurrent set of relative configurations.

First, $\L$ contains $\L_{\und{\om}_0}^0$\,, hence is non empty. It is compact by construction. Let $u = (\und{\te}, \und{\te}', s,t) \in \L$.

We can find $\und{\te}^0 \in \Delta_0$\,, $\und{\te}^{\prime 0} \in
\Delta_0'$\,, $t_0 \in \Delta_2$ with $d(\und{\te}, \und{\te}^0) < \ro^{5/2}$, \,\, $d(\und{\te}', \und{\te}^{\prime 0}) < \ro^{5/2}$, \,\, $|t-t_0| < \ro^{5/2}$. Then, by construction of $\widetilde\L$, we have still $(\und{\te}^0, \und{\te}^{\prime 0}, s) \in \widetilde\L$, hence we can find
$s_0 \in \Delta_1(\und{\te}^0, \und{\te}^{\prime 0})$ with $|s-s_0| <
\ro^{5/2}$. Let $u_0 = (\und{\te}^0, \und{\te}^{\prime 0}, s_0, t_0) \in \Delta$.

With $\und{\widetilde\te}$, $\und{\widetilde\te}'$, $\tilde s$, $\tilde t$ related to $\und{\te}$, $\und{\te}'$, $s$, $t$ by the definition of $\L$ above, we have
$$
\align
d(\und{\te}^0, \und{\widetilde\te}) & < 2\ro^{5/2},\\
d(\und{\te}^{\prime 0}, \und{\widetilde\te}') & < 2\ro^{5/2},\\
|s_0-\tilde s| &\le \ro/2 + \ro^{5/2} < \ro,\\
|t_0-\tilde t| &\le \ro/2 + \ro^{5/2} < \ro,
\endalign
$$
which means that $u_0 \in \L_{\und{\om}_0}^1$\,; in other words, $\und{\om}_0
\in \Om^1(u_0)$.

Then by choice of $\und{\om}_0$\,, we have $\und{\om}_0 \in \Om^0(u_0)$. This
means that there exists $\und{b} \in \Sigma(\ro)$, $\und{b}' \in \Sigma'(\ro)$,
with $b_0 = \te_0^0$\,, $b_0' = \te_0^{\prime 0}$ such that, if
$$
T_{\und{b}}^{\und{\om}_0}\,T_{\und{b}'}'(u_0) = (\und{\widehat\te}^0,
\und{\widehat\te}^{\prime 0}, \hat s_0, \hat t_0) = \hat u_0\,,
$$
we have
$$
(\widehat\te^0, \widehat\te^{\prime 0}, \hat s_0') \in \widetilde\L
\quad\text{for}\quad |\hat s_0 - \hat s_0'| < \frac 12 \ro^{1/2}
$$
and $\hat t_0 \in L_{\und{\om}_0}^0(\und{\widehat\te}^0,
\und{\widehat\te}^{\prime 0}, \hat s_0)$.

\noindent Let
$$
\hat u = T_{\und{b}}^{\und{\om}_0} T_{\und{b}'}'(u) = (\und{\widehat\te},
\und{\widehat\te}', \hat s, \hat t).
$$

\vglue .2in

\noindent{\bf Lemma}: We have
$$
\alignedat{2}
&d(\und{\widehat\te}, \und{\widehat\te}^0) \le c_9\,\ro^{7/2}, &&\qquad
d(\und{\widehat\te}', \und{\widehat\te}^{\prime 0}) \le c_9\,\ro^{7/2}\\
&|\hat s - \hat s_0| \le c_9\,\ro^{5/2}, &&\qquad |\hat t - \hat t_0| \le c_9\,\ro^{3/2}.
\endalignedat
$$

\vglue .1in

\noindent{\bf Proof}: By proposition \S1.3, the $C^1$-distance to the identity of $k^{\und{\te}} \circ (k^{\und{\te}^0})^{-1}$, \,\, $k^{\prime\und{\te}'} \circ (k^{\prime\und{\te}^{\prime 0}})^{-1}$ is at most of order $\ro^{5/2}$.
The estimates from the lemma then follow immediately from the definitions of $T_{\und{b}}$\,, $T_{\und{b}'}$\,. \qquad\qed

To prove that $\hat u \in \tni \L$, we take $(\widehat\te^1,
\widehat\te^{\prime 1}, \hat s_1, \hat t_1)$ with
$$
\alignedat{2}
&d(\und{\widehat\te}^1, \und{\widehat\te}) \le \ro^{7/2}, &&\qquad
d(\und{\widehat\te}^{\prime 1}, \und{\widehat\te}')\le \ro^{7/2}\\
&|\hat s_1 - \hat s| \le \ro^{5/2}, &&\qquad |\hat t_1 - \hat t| \le \ro^{3/2}.
\endalignedat
$$
We have then
$$
\alignedat{2}
&d(\und{\widehat\te}^1, \und{\widehat\te}^0) \le (1+c_9)\ro^{7/2}, &&\qquad
d(\und{\widehat\te}^{\prime 1}, \und{\widehat\te}^{\prime 0}) \le
(1+c_9)\ro^{7/2},\\
&|\hat s_1 - \hat s_0| \le (1+c_9)\ro^{5/2}, &&\qquad |\hat t_1 - \hat t_0| \le (1+c_9)\ro^{3/2}
\endalignedat
$$
which imply first $(\und{\widehat\te}^1, \und{\widehat\te}^{\prime 1}, \hat s_1) \in \widetilde\L$, and then $(\und{\widehat\te}^1,
\und{\widehat\te}^{\prime 1}, \hat s_1, \hat t_1) \in \L$.

\vglue .2in

\noindent{\bf 3.11}\quad In order to prove the theorem in \S2.5, one has to modify slightly the argument as follows.

Let $0 < \tilde d < d + d'- 1$; choose $\tilde d < d^* < d + d'- 1$. We will indicate at the end of \S4 how to define $\ov{L}_{\und{\om}}^0(\und{\te},\und{\te}',s)$ (analogous to $L_{\und{\om}}^0(\und{\te}, \und{\te}',s)$), for $(\und{\te}, \und{\te}',s) \in \widetilde\L$.

For $u \in \widetilde\L \times\re$, one defines $\ov\Om^1(u)$ as $\Om^1(u)$ above (using $\ov L_{\und{\om}}^0$ instead of $L_{\und{\om}}^0$). Then, $\ov\Om^0(u)$ is the set of $\und{\om} \in \Om$ such that, with $M = [\ro^{-d^*}]$, there exist pairs $(\und{b}^1, \und{b}^{'1}),\dots,(\und{b}^M, \und{b}^{'M}) \in \Sigma(\ro)\times\Sigma'(\ro)$ (with disjoint $I(\und{b}^i))$ for which properties (i), (ii) hold [we had $M=1$ in \S3.8].

We will see in section 5 that in this case we still have (as proposition 3.8)
$$
\bp(\ov\Om^1(u) - \ov\Om^0(u)) \le \exp(-\bar c_7\,\ro^{-\frac{1}{2k}(d+d'-1)}).
$$

The set $\ov\L_{\und{\om}_0}$\,, defined as in 3.10 with the $\ov L_{\und{\om}_0}^0$ instead of $L_{\und{\om}_0}^0$\,, is easily seen to be $\tilde d$-recurrent if $\und{\om}_0$ is selected as in 3.9.

\vglue .2in

\noindent{\bf 3.12 Remark}: Consider a smooth family $(g_\mu, g_\mu')$, $\mu \in [-1,+1]^r$ of maps as above such that one has $HD(K_\mu) + HD(K_\mu') > 1$ for all $\mu$.

We consider perturbations $(g_\mu^{\und{\om}}, g_\mu')$ constructed as above, where now $\mu$ is restricted to a $\ro^3$-dense finite subset $\Delta_3$ of the parameter space $[-1,1]^n$, with
$$
\#\,\Delta_3 \le c_n\,\ro^{-3n}.
$$

As this is polynomial in $\ro$, we can select $\und{\om}_0$ such that, for any $u \in \Delta$, $\mu \in \Delta_3$ we have either $\und{\om}_0 \notin \Om_\mu^1(u)$ or $\und{\om}_0 \in \Om_\mu^0(u)$.

For any $\mu_0 \in \Delta_3$\,, this allows to construct (for $(g_{\mu_0}^{\und{\om}_0}, g_{\mu_0}')$) a recurrent set $\L_{\mu_0}$\,.

But, in view of the construction of this set, it is fairly clear that $\L_{\mu_0}$ is still recurrent for $(g_\mu^{\und{\om}_0}, g_\mu')$, provided $||\mu-\mu_0|| < \ro^3$.

Thus the following reinforcement of the theorem in \S2.4 is true: there exists open and dense sets of families as above for which one can, for {\it all\/} values of the parameter, find a recurrent compact set of configurations.

Moreover, generic families have the following property: for each value of the parameter, there are periodic points of $g_\mu$\,, $g_\mu'$ respectively such that the ratio of the logarithms of the moduli of the corresponding eigenvalues is irrational.

This allows (with the same proof) the following reinforcement of the second theorem in \S2.4: the open set $V \subset \Om_\Sigma \times \Om_{\Sigma'}$ defined in \S2.4 is not only dense in $\{(g,g'), HD(K) + HD(K') > 1\}$, but its complement is infinite codimensional in the sense that generic $n$-parameter families (for any $n$) take their values in $V$.

\vglue .5in

\noindent{\bf \S4.\quad Application of Marstrand's theorem: construction of the
sets $\pmb{E(\und{a}, \und{a}')}$ and $\pmb{L_{\und{\om}}^0(\und{\te}, \und{\te}',s)}$.}

\vglue .2in

\noindent{\bf 4.1}\quad Given a point $(\und{\te}, \und{\te}',s)$ in the space
$\si$ of relative scales, and points $x \in K(\te_0)$, $x' \in K'(\te_0')$,
there is a unique relative configuration above $(\und{\te}, \und{\te}',s)$ such
that
$$
A(k^{\und{\te}}(x)) = A'(k^{\prime\und{\te}'}(x')),
$$
(where $(\und{\te},A), (\und{\te}',A')$ represents this configuration).
Parametrizing the fiber of $\C$ over $(\und{\te}, \und{\te}',s)$ as in \S3.3,
the coordinate of this configuration is given by
$$
\align
t &:= \pi_{\und{\te},\und{\te}',s}(x,x')\\
&= k^{\prime\und{\te}'}(x') - k^{\prime\und{\te}'}(\om(\te_0')) -
s(k^{\und{\te}}(x) - k^{\und{\te}}(\om(\te_0))).
\endalign
$$

\vglue .2in

\noindent{\bf 4.2}\quad We equip each set $K(\te_0)$ (resp. $K'(\te_0'))$ with
the $d$-dimensional (resp. $d'$-dimensional) Hausdorff measure $\mu_d$ (resp.
$\mu_{d'}$).

Then, for $(\und{\te},\und{\te}',s) \in \si$, we denote by
$\mu(\und{\te},\und{\te}',s)$ the image under $\pi_{\und{\te},\und{\te}',s}$ of
$\mu_d \times \mu_{d'}$ on $K(\te_0) \times K'(\te_0')$.

It is well known that these exists constants $c_{11} > c_{10} > 0$ such that,
for $\te_0 \in \AA$, $\te_0' \in \AA'$:
$$
c_{10} < \mu_d \times \mu_{d'}(K(\te_0) \times K'(\te_0')) < c_{11}\,.
$$

Now, Kaufman's proof of Marstrand's theorem (see [PT2]) tells us that, for
fixed $(\und{\te}, \und{\te}')$, the measure $\mu(\und{\te},\und{\te}',s)$ is absolutely continuous with respect to
Lebesgue measure for Lebesgue almost every $s$, with $L^2$-density
$\chi_{\und{\te},\und{\te}',s}$ satisfying
$$
\int_{J_R} \lV \chi_{\und{\te},\und{\te}',s}\rV_{L^2}^2\,ds \le c_{12}(R),
$$
where $c_{12}(R)$ is independent on $\und{\te}$, $\und{\te}'$.

\vglue .2in

\noindent{\bf 4.3}\quad When one does control $\lV
\chi_{\und{\te},\und{\te}',s}\rV_{L^2}$\,, this gives, by Cauchy-Schwarz
inequality, a lower bound for the Lebesgue measure of
$\pi_{\und{\te},\und{\te}',s}(X)$,\,\, $X$ being a subset of $K \times K'$ with
positive $(d+d')$-dimensional Hausdorff measure; indeed we have:
$$
\align
\mu_d\times\mu_{d'}(X) &\le \int_{\pi_{\und{\te},\und{\te}',s}(X)}
\chi_{\und{\te},\und{\te}',s}(t)\,dt\\
&\le \lv\pi_{\und{\te},\und{\te}',s}(X)\rv^{1/2} 
\lV \chi_{\und{\te},\und{\te}',s}\rV_{L^2}
\endalign
$$
and therefore
$$
\lv\pi_{\und{\te},\und{\te}',s}(X)\rv \ge (\mu_d\times\mu_{d'}(X))^2 
\lV \chi_{\und{\te},\und{\te}',s}\rV_{L^2}^{-2}\,.
$$

\vglue .2in

\noindent{\bf 4.4}\quad Fix $(\und{\te},\und{\te}')$ in $\Sigma^- \times
\Sigma^{\prime -}$. Let $\und{a} \in \Sigma(\ro^{1/2k})$, $a' \in
\Sigma'(\ro^{1/2k})$, with $a_0 = \te_0$\,, $a_0' = \te_0'$\,. as seen in 3.1, one has
$$
T_{\und{a}} T_{\und{a}'}'(\und{\te},\und{\te}',s) = (\und{\te}a_1\dots a_n, \und{\te}' a_1\dots a_{n'}\,,
\ve\ve's|I^{\und{\te}}(\und{a})|\,|I^{\und{\te}'}(\und{a}')|^{-1})
$$
where $\ve,\ve' \in \{\pm 1\}$ and
$$
c_{13}^{-1} \le |I^{\und{\te}}(\und{a})|\,|I^{\und{\te}'}(\und{a}')|^{-1} \le
c_{13}\,.
$$

We therefore have
$$
\int_{J_R} \lV \chi\_{T_{\und{a}}
T_{\und{a}'}'(\und{\te},\und{\te}',s)}\rV_{L^2}^2 \,ds \le c_{12}'(R),
$$
with $c_{12}'(R)$ independent of $\und{\te}$, $\und{\te}'$, $\und{a}$,
$\und{a}'$. On the other hand, one has
$$
\align
&\#\,\Sigma(\ro^{1/2k}) \le c_{14}\,\ro^{-d/2k}\,,\\
&\#\, \Sigma'(\ro^{1/2k}) \le c_{14}\,\ro^{-d'/2k}\,.
\endalign
$$
We conclude that
$$
\int_{J_R} \sum_{\und{a},\und{a}'} \lV \chi\_{T_{\und{a}}
T_{\und{a}'}'(\und{\te},\und{\te}',s)}\rV_{L^2}^2 \,ds \le
c_{14}^2\,\ro^{-\frac{d+d'}{2k}}\, c_{12}'(R).
$$

We now define, with $c_{15} > 0$ conveniently large to be determined later:
$$
\align
E(\und{\te},\und{\te}') &= \bigg\{s \in J_R\,, \lV
\chi_{\und{\te},\und{\te}',s}\rV_{L^2}^2  \le c_{15}\\
&\quad\text{and }\quad \sum_{\und{a},\und{a}'} \lV \chi\_{T_{\und{a}}
T_{\und{a}'}'(\und{\te},\und{\te}',s)}\rV_{L^2}^2 \le c_{15}\,\ro^{-
\frac{d+d'}{2k}}\bigg\}.
\endalign
$$

For $\und{c} \in \Sigma(\ro^{1/2})$, $\und{c}' \in \Sigma'(\ro^{1/2})$, we
define $E(\und{c}, \und{c}')$ as the set of $s \in J_R$ such that there exists
$\und{\te}$, $\und{\te}'$ ending respectively with $\und{c}$, $\und{c}'$ such
that $s \in E(\und{\te}, \und{\te}')$.

One has, for any $\und{\te} \in \Sigma^-$, $\und{\te}' \in \Sigma^{\prime -}$:
$$
m(J_R - E(\und{\te},\und{\te}')) \le c_{15}^{-1}(c_{12}(R) +
c_{14}^2\,c_{12}'(R));
$$
therefore, provided that
$$
c_{15} > c_1^{-1}(c_{12}(R) + c_{14}^2\,c_{12}'(R)),
$$
we will have
$$
m(J_R - E(\und{c},\und{c}')) \le c_1
$$
for all $\und{c} \in \Sigma(\ro^{1/2})$, $\und{c}' \in \Sigma'(\ro^{1/2})$. 
This means that we can apply the Scale recurrence Lemma with the family
$E(\und{c},\und{c}')$ of subsets of $J_R$\,.

Let $E^*(\und{c},\und{c}')$ be then the family of subsets of $J_R$ given by the
Scale recurrence Lemma (cf. 3.7). We know that each $E^*(\und{c},\und{c}')$ is
contained in a $c_2\,\ro^{1/2}$-neighbourhood of $E(\und{c},\und{c}')$.

\vglue .2in

\noindent{\bf 4.5}\quad Until now, we have only worked with the unperturbed maps $g$, $g'$ (except from a possible initial perturbation, see \S 3.4). We now consider the family of random perturbations $g^{\und{\om}}$ of \S3.5, and will proceed to construct the sets $L_{\und{\om}}^0(\und{\te},\und{\te}',s)$,
for $(\und{\te},\und{\te}',s) \in \widetilde\L$. For $\und{a} \in
\Sigma(\ro^{1/2k})$, let $\Sigma^-(\und{a})$ be the open and closed subset of $\Sigma^-$ formed by the $\und{\te}$ ending with $\und{a}$. Choose a subset $\Sigma_2^-$ of $\Sigma(\ro^{1/2k})$ such that
$$
\Sigma^- = \bigcup_{\Sigma_2^-} \Sigma^-(\und{a})
$$
is a partition of $\Sigma^-$.

For $\und{a} \in \Sigma_2^-$\,, define a subset $\Sigma_1(\und{a})$ of the subset $\Sigma_1$ of \S3.5 (recall $\Sigma_1 \subset \Sigma(\ro^{1/k})$), as the set of words in $\Sigma_1$ starting with $\und{a}$. For $\und{\te} \in \Sigma^-(\und{a})$, we also define $\Sigma_1(\und{\te}) = \Sigma_1(\und{a})$.

Let $\und{\te} \in \Sigma^-$. We write
$$
\align
\Om &= [-1,+1]^{\Sigma_1(\und{\te})} \times [-1,+1]^{\Sigma_1-
\Sigma_1(\und{\te})},\\
\und{\om} &= (\und{\om}',\und{\om}'')\\
\intertext{and for such an $\und{\om}$, we set}
\und{\om}^* &= (\und{0},\und{\om}'').
\endalign
$$
This depends on $\und{\te}$, but nearby $\und{\widehat\te}$ (with
$d(\und{\te}, \und{\widehat\te}) < c_0^{-1}\,\ro^{1/2k}$) will belong to the same $\Sigma^-(\und{a})$ and give the same projection $\und{\om}^*$ of $\und{\om}$.

For $(\und{\te},\und{\te}',s) \in \widetilde\L$, the set
$L_{\und{\om}}^0(\und{\te},\und{\te}',s)$ will actually only depend (as far as $\und{\om}$ is concerned) on the projection $\und{\om}^*$ of $\und{\om}$ associated to $\und{\te}$.

We will say that two words $\und{b}^0,\und{b}^1 \in \Sigma(\ro)$ are {\it independent\/} if there is no word $\und{b} \in \Sigma(\ro^{1/2k})$ such that both $\und{b}^0$ and $\und{b}^1$ start with $\und{b}$.

With $c_{16} > 0$ conveniently small, to be chosen in the following, let
$$
N = \bigg[c_{16}^2\,\ro^{-\frac{1}{2k}(d+d'-1)}\bigg].
$$

\noindent{\bf 4.6}\quad Let $(\und{\te},\und{\te}',s) \in \widetilde\L$, and $\und{\om} \in \Om$.

We define $L_{\und{\om}}^0(\und{\te},\und{\te}',s)$ to be the set of points $(\und{\te},\und{\te}',s,t)$ in the fiber for which there exist pairs $(\und{b}^1,\und{b}^{\prime 1}),\dots,(\und{b}^N,\und{b}^{\prime N})$ in $\Sigma(\ro) \times \Sigma'(\ro)$, with $b_0^i = \te_0$\,,\,\, $b_0^{\prime i} = \te_0'$ such that, if we set
$$
T_{\und{b}^i}^{\und{\om}^*}\,T_{\und{b}^{\prime i}}'(\und{\te},\und{\te}',s,t)
= (\und{\te}^i, \und{\te}^{\prime i}, s_i,t_i),
$$
the following hold:

\item{(i)} the words $\und{b}^1,\dots,\und{b}^N$ are pairwise independent;
\item{(ii)} for $1 \le i \le N$, \,\,\, $\und{\te}^i \in \Sigma_{nr}^-$\,;
\item{(iii)} for $1 \le i \le N$, and $|\tilde s - s_i| \le \dfrac 23
\ro^{1/2}$,\,\, $(\und{\te}^i, \und{\te}^{\prime i}, \tilde s) \in
\widetilde\L$;
\item{(iv)} for $1 \le i \le N$,\,\,\, $|t_i| \le 2(1+R)$.

We will use also a slightly smaller set $L_{\und{\om}}^{-
1}(\und{\te},\und{\te}',s)$; it is defined in the same way than
$L_{\und{\om}}^0(\und{\te},\und{\te}',s)$, but with (iii), (iv) replaced by:

\item{(iii)'} for $1 \le i \le N$, and $|\tilde s-s_i| \le \dfrac 34
\ro^{1/2}$, $(\und{\te}^i, \und{\te}^{\prime i}, \tilde s) \in \widetilde\L$
\item{(iv)'} for $1 \le i \le N$, \,\, $|t_i| \le 1+R$.

In the end of this section, we will prove the following estimate.

\vglue .2in

\noindent{\bf Proposition}: If $c_{16}$ has been chosen sufficiently small, there exists $c_{17} > 0$ such that, for any $(\und{\te}, \und{\te}',s) \in \widetilde\L$ and any $\und{\om} \in \Om$, the Lebesgue measure of $L_{\und{\om}}^{-1}(\und{\te},\und{\te}',s)$  is $> c_{17}$\,.

\comment
\vglue .2in

\noindent{\bf 4.6.1}\quad Let $0 < \tilde d < d+d'-1$,\,\, $d^* = \dfrac 12 (\tilde d+d+d'-1)$, \,\, $M = [\ro^{-d^*}]$. 
We define $\ov{L}_{\und{\om}}^0(\und{\te},\und{\te}',s)$ to be the set of points $(\und{\te}, \und{\te}',s,t)$ in the fiber for which there exist pairs $(\und{b}^{ij}, \und{b}^{\prime ij})$ with all the $\und{b}^{ij}$ distinct, $1 \le i \le M$,\,\, $1 \le j'\le N$ with $b_0^{ij} = \te_0$\,,\,\, $b_0^{\prime ij} = \te_0'$ such that, if we set $T_{\und{b}^{ij}}^{\und{\om}^*}\,T_{\und{b}^{\prime ij}}' (\und{\te}, \und{\te}',s,t) = (\und{\te}^{ij}, \und{\te}^{\prime ij}, s_{ij},t_{ij})$, the conditions i)-iv) as before hold. We define $\ov{L}_{\und{\om}}^{-1}(\und{\te},\und{\te}',s)$ in an analogous way.

In the end of this section we will obtain the same estimate of 4.5 to $\ov L_{\und{\om}}^{-1}$ instead of $L_{\und{\om}}^{-1}$. This will show that for a fixed $i$,\,\, $1 \le i \le M$, the probability of no one of the $(\und{\te}^{ij}, \und{\te}^{\prime ij}, s_{ij}, t_{ij})$ return to $\ov{L}_{\und{\om}}^0$ is at most $\exp(-\ov c_7 \ro^{-1/2k(d+d'-1)})$, which implies that the probability of the existence of some $i$,\,\, $1 \le i \le M$ as above is at most $M\cdot\exp(-\ov c_7 \ro^{-\frac{1}{2k}(d+d'-1)})$, implying the statement of 3.8.1.
\endcomment

\vglue .2in

\noindent{\bf 4.7}\quad In the proof of the proposition, we will use the following estimates on the family of random perturbations, to be proved in the next section.

\vglue .2in

\noindent{\bf Lemma}: Let $\und{\te} \in \Sigma^-$,\,\, $\und{\om} \in \Om$.

\item{(i)} for $\und{a} \in \Sigma^+$, with $a_0=\te_0$\,, we have
$$
|k^{\und{\te},\und{\om}}(h^{\und{\om}}(\und{a})) - k^{\und{\te}}(h(\und{a}))|
\le c_{18}c_5\,\ro^{1-\frac{1}{2k}};
$$

\item{(ii)} for $\und{a}$ a word with $a_0=\te_0$\,, \,\, $|I(\und{a})| >
c_0^{-1}\,\ro$, we have
$$
\lv \Log
\frac{|I^{\und{\te},\und{\om}(\und{a})|}}{|I^{\und{\te}(\und{a})}|}\rv
\le c_{18}c_5\,\ro^{1-\frac{1}{2k}}.
$$

The constant $c_{18}$ is independent of $\und{\te}$, $\und{\om}$, $\und{a}$,
$\ro$, and the size $c_5$ of the perturbation.

\vglue .2in

\noindent{\bf 4.8}\quad Let $(\und{\te},\und{\te}',s) \in \widetilde\L$ be
given.

Choose a subfamily $\Sigma_2$ of $\Sigma(\ro^{1/2k})$ of words starting with
$\te_0$ such that
$$
K(\te_0) = \bigcup_{\Sigma_2} K(\und{a})
$$
is a partition of $K(\te_0)$. Similarly, choose a subfamily $\Sigma_2'$ of
$\Sigma'(\ro^{1/2k})$ of words starting with $\te_0'$ such that
$$
K'(\te_0') = \bigcup_{\Sigma_2'} K'(\und{a}').
$$
For each $(\und{a},\und{a}') \in \Sigma_2 \times \Sigma_2'$\,, we have
$$
c_{19}^{-1}\, \ro^{\frac{1}{2k}(d+d')} \le \mu_d \times \mu_{d'}(K(\und{a})
\times K'(\und{a}')) \le c_{19}\, \ro^{\frac{1}{2k}(d+d')}\,.
$$

Let $J(\und{a},\und{a}') := \pi_{\und{\te},\und{\te}',s}(I(\und{a}) \times
I(\und{a}'))$ for $\und{a} \in \Sigma_2$\,, $\und{a}' \in \Sigma_2$\,; we have
$$
c_{19}^{-1}\,\ro^{\frac{1}{2k}} \le |J(a,a')| \le c_{19}\,
\ro^{\frac{1}{2k}}\,.
$$
Say $(\und{a},\und{a}')$ is {\it good\/} if there are no more than $c_{16}^{-
1}\, \ro^{-1/2k(d+d'-1)}$ intervals $J(\und{\tilde a},\und{\tilde a}')$ whose
centers are distant from the center of $J(\und{a},\und{a}')$ by less than
$2c_{19}\,\ro^{1/2k}$.

\vglue .2in

\noindent{\bf Lemma}: The number of bad pairs $(\und{a},\und{a}')$ is less than $12c_{19}^3\, c_{16}c_{15}\, \ro^{-1/2k(d+d')}$.

\vglue .1in

\noindent{\bf Proof of lemma}: By construction of $\widetilde\L$, there exists $\und{\widetilde\te}$, $\und{\widetilde\te}'$, $\tilde s$ with $d(\und{\te},\und{\widetilde\te}) \le (c_2+c_0)\ro^{1/2}$, $d(\und{\te}',
\und{\widetilde\te}') \le (c_2+c_0)\ro^{1/2}$, $|s-\tilde s| \le
c_2\,\ro^{1/2}$ such that
$$
\lV \chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}\rV_{L^2}^2 \le
c_{15}\,.
$$

Therefore, if we replace $\und{\te}$, $\und{\te}'$, $s$ by
$\und{\widetilde\te}$, $\und{\widetilde\te}'$, $\tilde s$ the endpoints of $J(\und{a},\und{a}')$ only move by a distance of order at most $\ro^{1/2}$. It is thus sufficient to prove the lemma for $\und{\widetilde\te}$, $\und{\widetilde\te}'$, $\tilde s$ with $2c_{19}\,\ro^{1/2k}$ replaced by $5/2\,c_{19}\,\ro^{1/2k}$.

Let $(\und{a},\und{a}')$ be bad; then there are at least $c_{16}^{-1}\,\ro^{-1/2k(d+d'-1)}$ pairs $(\tilde a, \tilde a')$ such that $J(\tilde a, \tilde a')$ is contained in the interval $J^*(\und{a},\und{a}')$ of same center than
$J(\und{a},\und{a}')$ and length $6\,c_{19}\,\ro^{1/2k}$. This means that
$$
\int_{J^*(\und{a},\und{a}')}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s} \ge c_{19}^{-
1}\,c_{16}^{-1}\,\ro^{1/2k} = \frac 16 c_{19}^{-2}\, c_{16}^{-1}
|J^*(\und{a},\und{a}')|.
$$
But then, by Cauchy-Schwarz
$$
\align
\frac 16 c_{19}^{-2}\, c_{16}^{-1}|J^*(\und{a},\und{a}')|
\int_{J^*(\und{a},\und{a}')}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s} &\le
\bigg(\int_{J^*(\und{a},\und{a}')}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}\bigg)^2\\
&\le |J^*(a,a')| \int_{J^*(a,a')}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}^2
\endalign
$$
and thus
$$
\int_{J^*(\und{a},\und{a}')}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}^2 \ge \frac 16
c_{19}^{-2}\,c_{16}^{-1} \int_{J^*(\und{a},\und{a}')}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}
$$
Let $J^*$ be the union over all bad pairs $(\und{a},\und{a}')$ of the
intervals $J^*(\und{a},\und{a}')$; one can extract a subfamily of intervals
whose union is $J^*$ and which does not cover any point more than twice.

Therefore we obtain
$$
\int_{J^*} \chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}^2 \ge
\frac{1}{12} c_{19}^{-2}\,c_{16}^{-1} \int_{J^*}
\chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}.
$$
As $\lV \chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}\rV_{L^2}^2 \le
c_{15}$\,, we get
$$
\int_{J^*} \chi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s} \le
12\,c_{19}^2\,c_{16}\,c_{15}\,.
$$
As $J^*$ contains $J(\und{a},\und{a}')$ for all bad $(\und{a},\und{a}')$, the
estimate of the lemma follows. \qquad\qed

\vglue .2in

\noindent{\bf 4.9}\quad To keep the argument clearer, we now prove Proposition
4.6 when $\und{\om}=0$ (for the unperturbed map $g$). Afterwards we will
indicate how to modify the argument for general $\und{\om}$.

First, we construct the pairs $(\und{b},\und{b}')$ amongst which the pairs $(\und{b}^i, \und{b}^{\prime i})$ of
4.6 must be looked for.
We make the following easy observation:

\vglue .2in

\noindent{\bf Lemma}: Let $\und{\te} \in \Sigma_{nr}^-$\,. The number of words $\und{c} \in \Sigma(\ro^{1/2})$ with $c_0=\te_0$\,, such that $\und{\te}c_1\dots c_m \notin \Sigma_{nr}^-$ is $o(\ro^{-1/2d})$ as $\ro\to0$, uniformly in $\und{\te}$.

Now let $(\und{\te},\und{\te}',s) \in \widetilde\L$. It follows from conclusion (ii) of the Scale recurrence Lemma and the last observation that we can find at least $\dfrac 12 c_3\,\ro^{-1/2(d+d')}$ pairs $(\und{c}^i, \und{c}^{\prime i}) \in
\Sigma(\ro^{1/2}) \times \Sigma'(\ro^{1/2})$ such that, writing $T_{\und{c}^i} T_{\und{c}^{\prime i}}'(\und{\te},\und{\te}',s) = (\und{\te}^i,\und{\te}^{\prime i},s_i)$, we have:

\item{--} $\und{\te}^i \in \Sigma_{nr}^-$\,;
\item{--} $[s_i-\ro^{1/2}, s_i+\ro^{1/2}] \subset E^*(\und{c}^i,
\und{c}^{\prime i})$.

\noindent As $(\und{\te}^i, \und{\te}^{\prime i},s_i)$ again belongs to $\widetilde\L$, we can for each $i$ find at least $\dfrac 12 c_3\,\ro^{-1/2(d+d')}$ pairs $(\und{d}^{ij}, \und{d}^{\prime ij}) \in \Sigma(\ro^{1/2}) \times \Sigma'(\ro^{1/2})$ (with the first letter of $\und{d}^{ij}$, $\und{d}^{\prime ij}$ being the last one of $\und{c}^i$, $\und{c}^{\prime i}$ respectively), such that writing $T_{\und{d}^{ij}} T_{\und{d}_{ij}'}'(\und{\te}^i,\und{\te}^{\prime i},s_i) = (\und{\te}^{ij}, \und{\te}^{\prime ij}, s_{ij})$, we have

\item{--} $\und{\te}^{ij} \in \Sigma_{nr}^-$\,;
\item{--} $[s_{ij}-\ro^{1/2}, s_{ij}+\ro^{1/2}] \in E^*(\und{d}^{ij},
\und{d}^{\prime ij})$.

\noindent Concatenation of the $\und{c}^i$, $\und{c}^{\prime i}$ and 
$\und{d}^{ij}, \und{d}^{\prime ij}$ gives a family of words
$(\und{b}^{ij},\und{b}^{\prime ij})$ in $\Sigma(\ro) \times \Sigma'(\ro)$ with at least $\dfrac 14 c_3^2\,\ro^{-(d+d')}$ elements.

Let $(\und{\widetilde\te},\und{\widetilde\te}',\tilde s)$ be such that
$d(\und{\te},\widetilde\te) \le (c_0+c_2)\ro^{1/2}$,\,\,
$d(\und{\te}',\und{\widetilde\te}') \le (c_0+c_2)\ro^{1/2}$,\,\, $|s-\tilde s|\le c_2\,\ro^{1/2}$ and  $\tilde s \in E(\widetilde\te, \widetilde\te')$.

\vglue .2in

\noindent{\bf Lemma}: If $c_{16}$ has been chosen sufficiently small, there are at least $\dfrac 16 c_3\,c_{19}^{-2}\, \ro^{-\frac{d+d'}{2k}}$ pairs $(\und{a},\und{a}') \in \Sigma_2 \times \Sigma_2'$ which are good and satisfy
$$
\lV \chi_{T_{\und{a}}T_{\und{a}'}'(\widetilde\te, \widetilde\te',\tilde s)}\rV_{L^2}^2 \le c_{16}^{-1}
$$
and such that at least $\dfrac 16 c_3c_{19}^{-1}\,\ro^{-(d+d')(\frac 12-\frac{1}{2k})}$ pairs $(\und{c}^i, \und{c}^{\prime i})$ start with $(\und{a}, \und{a}')$.

\vglue .1in

\noindent{\bf Proof}: By definition of $E(\und{\widetilde\te},
\und{\widetilde\te}')$, the number of pairs $(\und{a},\und{a}') \in \Sigma_2 \times \Sigma_2'$ such that
$$
\lV \chi_{T_{\und{a}}T_{\und{a}'}'(\widetilde\te, \widetilde\te',\tilde
s)}\rV_{L^2}^2 > c_{16}^{-1}
$$
is at most $c_{16}c_{15}\,\ro^{-\frac{1}{2k}(d+d')}$. Call $(\und{a},\und{a}')$ very good if it is good and satisfy
$$
\lV \chi_{T_{\und{a}}T_{\und{a}'}'(\widetilde\te, \widetilde\te',\tilde
s)}\rV_{L^2}^2 \le c_{16}^{-1}
$$
The number of pairs $(\und{a},\und{a}')$ which are not very good is thus $\le c_{15}c_{16}(1+12c_{19}^3)\ro^{-\frac{1}{2k}(d+d')}$.

But for each $(\und{a},\und{a}') \in \Sigma_2 \times \Sigma_2'$\,, the number of pairs $(\und{c},\und{c}') \in \Sigma(\ro^{1/2}) \times \Sigma'(\ro^{1/2})$ starting with $(\und{a},\und{a}')$ is at most
$c_{19}^2\,\ro^{(d+d')(\frac{1}{2k}-\frac 12)}$.

It follows, if $c_{16}$ is small enough, that at least $\dfrac 13 c_3\,\ro^{-1/2(d+d')}$ pairs $(\und{c}^i,\und{c}^{\prime i})$ start with a very good $(\und{a},\und{a}') \in \Sigma_2\times\Sigma_2'$\,.

The total number of pairs $(\und{a},\und{a}')$ is at most $c_{19}\,\ro^{-\frac{d+d'}{2k}}$; if we discard those pairs for which less than $\dfrac 16 c_3c_{19}^{-1}\,\ro^{-(d+d')(\frac 12-\frac{1}{2k})}$ pairs $(\und{c}^i,\und{c}^{\prime i})$ begin with $(\und{a},\und{a}')$, we are still left with $\dfrac 16 c_3\,\ro^{-1/2(d+d')}$ pairs $(\und{c}^i, \und{c}^{\prime
i})$, of which no more than $c_{19}^2\,\ro^{-(d+d')(\frac 12-\frac{1}{2k})}$ may begin with the same $(\und{a},\und{a}')$. \qquad\qed

\vglue .2in

\noindent{\bf 4.10}\quad Let us call excellent those pairs $(\und{a},\und{a}')$ given by the last lemma.

Recall that we have defined
$$
J(\und{a},\und{a}') = \pi_{\und{\te},\und{\te}',s}(I(\und{a})\times
I(\und{a}')),
$$
and define similarly $J(\und{c}^i, \und{c}^{\prime i})$, $J(\und{b}^{ij},
\und{b}^{\prime ij})$. We only consider those $i$ for which $(\und{c}^i,
\und{c}^{\prime i})$ starts with an excellent $(\und{a}, \und{a}')$.

The following general lemmas will be used repeteadly:

\vglue .2in

\noindent{\bf Lemma 1}: Let $(J_\al)\_{\al\in A}$\,, $(J_\al')\_{\al\in A}$ be
two finite families of intervals, and let $\la > 1$, $\ve > 0$, $\nu > 0$.

Assume that for every $\al$, we have:

\item{(i)} $\ve < |J_\al| < \la\ve$,\,\, $\ve < |J_\al'|$;
\item{(ii)} the centers of $J_\al$\,, $J_\al'$ are distant by no more than $\nu\ve$.

Then
$$
m\bigg(\bigcup_A J_\al'\bigg) \ge \frac{1}{\la(4\nu+4)} m\bigg(\bigcup_A
J_\al\bigg).
$$

\vglue .1in

\noindent{\bf Proof}: Let $A' \subset A$ be a minimal family with
$$
\bigcup_{A'} J_\al = \bigcup_A J_\al\,.
$$
Then each point in $\underset{A}\to{\bigcup}\,J_\al$ belongs at most to two
intervals $J_\al$\,, $\al \in A'$. Replace each $J_\al'$\,, $\al \in A'$, by
the interval $\widetilde J_\al'$ with same center and length $\ve$. One has
$$
\sum_{A'} |\widetilde J_\al'| \ge \la^{-1} \sum_{A'} |J_\al| \ge \la^{-
1}|\bigcup_A J_\al|.
$$

On the other hand, let $x\in\re$; if $x \in \widetilde J_\al'$\,, for $\al \in
A'$, then the center of $\widetilde J_\al'$ belongs to $\big[x -
\dfrac{\ve}{2}, x + \dfrac{\ve}{2}\big]$ and the center of $J_\al$ is contained
in $\big[x-\big(\nu+\dfrac 12\big)\ve, x+\big(\nu+\dfrac 12\big)\ve\big]$.
Replacing the $J_\al$ by intervals $\widetilde J_\al$ with same center and
length $\ve$, we see that there are at most $(4\nu+4)\quad\al \in A'$ such
that $x \in \widetilde J_\al'$\,. Then we have:
$$
\lv\bigcup_A J_\al'\rv \ge \lv\bigcup_{A'} \widetilde J_\al'\rv \ge
\frac{1}{4\nu+4} \Sigma|\widetilde J_\al'|. \qquad\qed
$$

\vglue .2in

\noindent{\bf Lemma 2}: Let $(J_\al)\_{\al\in A}$ be a finite family of intervals, and $\la > 1$. For each $\al$, let $K_\al$ be a subset of $J_\al$ with $m(K_\al) \ge \la^{-1}|J_\al|$. Then
$$
m\bigg(\bigcup_A K_\al\bigg) \ge \frac{\la^{-1}}{2} m\bigg(\bigcup_A
J_\al\bigg).
$$

\vglue .1in

\noindent{\bf Proof}: We extract a subfamily $A'$ such that
$\underset{A'}\to{\bigcup}\,J_\al = \underset{A}\to{\bigcup}\,J_\al$ and each
$x$ belongs to at most two $J_\al$\,, $\al \in A'$. Then we have
$$
\align
m\bigg(\bigcup_A K_\al\bigg) \ge m\bigg(\bigcup_{A'} K_\al\bigg) &\ge \frac 12
\sum_{A'} m(K_\al)\\
&\ge \frac{\la^{-1}}{2} \sum_{A'} |J_\al|. \qquad\qed
\endalign
$$


\vglue .2in

\noindent{\bf 4.11}\quad Now let $(\und{a},\und{a}')$ be an excellent pair. For $i$ such that $(\und{c}^i, \und{c}^{\prime i})$ starts with $(\und{a},\und{a}')$, let $\widetilde J(\und{c}^i,\und{c}^{\prime i}) =
\pi_{\und{\widetilde\te},\und{\widetilde\te}',\tilde s}(I(\und{c}^i) \times
I(\und{c}^{\prime i}))$, and let $\widetilde J_1(\und{a},\und{a}')$ be the
union of those $\widetilde J(\und{c}^i,\und{c}^{\prime i})$. The
$\mu_d\times\mu_{d'}$-measure of the union of the $K(\und{c}^i) \times
K'(\und{c}^{\prime i})$ is $\ge c_{20}\,\ro^{-\frac{1}{2k}(d+d')}$ (because $(\und{a}, \und{a}')$ is excellent), and follows then from
4.3 that
$$
|\widetilde J_1(\und{a},\und{a}')| \ge c_{21}\,c_{16}\,\ro^{-1/2k}.
$$
When we come back to $(\und{\te},\und{\te}',s)$ and define
$J_1(\und{a},\und{a}')$ to be the union of the $J(\und{c}^i,\und{c}^{\prime
i})$ starting with $(\und{a},\und{a}')$, the sizes of the
$J(\und{c}^i,\und{c}^{\prime i})$ are still of order $\ro^{1/2}$ and the
centers are distant from those of $\widetilde J(\und{c}^i,\und{c}^{\prime i})$
by distances of order at most $\ro^{1/2}$. Then, by Lemma 1 above, we get
$$
|J_1(\und{a},\und{a}')| \ge c_{22}^{-1}\,c_{16}\,\ro^{1/2k}.
$$

Similarly, for each $(\und{c}^i,\und{c}^{\prime i})$ (starting with
$(\und{a},\und{a}')$), let $J_1(\und{c}^i,\und{c}^{\prime i})$ be the union of the $J(\und{b}^{ij}, \und{b}^{\prime ij})$ starting with
$(\und{c}^i,\und{c}^{\prime i})$. Then are at least $\dfrac 12 c_3\,\ro^{-1/2(d+d')}$ such $(\und{b}^{ij}, \und{b}^{\prime ij})$, and
$(\und{\te}^i,\und{\te}^{\prime i},s_i) = T_{\und{c}^i}T_{\und{c}^{\prime
i}}'(\und{\te},\und{\te}',s)$ belongs to $\widetilde\L$.

Therefore, using again 4.3, we conclude that
$$
|J_1(\und{c}^i,\und{c}^{\prime i})| \ge c_{23}^{-1}|J(\und{c}^i,\und{c}^{\prime
i})|.
$$

[The argument is the same than above: we first consider
$(\und{\widetilde\te}^i, \und{\widetilde\te}^{\prime i},\tilde s_i)$, with
$d(\und{\te}^i,\und{\widetilde\te}^i)$, $d(\und{\te}^{\prime
i},\und{\widetilde\te}^{\prime i})$, $|s_i-\tilde s_i|$ of order $\ro^{1/2}$,
such that $\lV\chi_{\und{\widetilde\te}^i, \und{\widetilde\te}^{\prime
i},\tilde s_i}\rV_{L^2}^2 \le c_{15}$].

Finally, let $J_2(\und{a},\und{a}')$ be the union over those
$(\und{c}^i,\und{c}^{\prime i})$ starting with $(\und{a},\und{a}')$ of the sets
$J_1(\und{c}^i,\und{c}^{\prime i})$. By Lemma 2 above, we obtain that
$$
|J_2(\und{a},\und{a}')| \ge \frac 12 c_{23}^{-1} \,c_{22}^{-
1}\,c_{16}\,\ro^{1/2k}.
$$
Let $\vr$ (resp. $\vr_2)$ be the sum, over excellent pairs $(\und{a},\und{a}')$,
of the characteristic functions of $J(\und{a},\und{a}')$ (resp.
$J_2(\und{a},\und{a}'))$.

The number of excellent pairs is $\ge \dfrac 16 c_3c_{19}^{-2}\,\ro^{-
\frac{d+d'}{2k}}$ by 4.8, and therefore
$$
\int\vr_2 \ge \frac{1}{12} c_{23}^{-1}\,c_{22}^{-
1}\,c_3\,c_{19}^2\,c_{16}\,\ro^{-\frac{1}{2k}(d+d'-1)}.
$$
On the other hands, because excellent pairs are good, one has
$$
\vr_2 \le \vr \le c_{16}^{-1}\,\ro^{-1/2k(d+d'-1)}.
$$
We conclude that
$$
m\big(\big\{\vr_2 \ge c_{16}^2\,\ro^{-\frac{1}{2k}(d+d'-1)}\big\}\big) \ge
c_{24}\,c_{16}^2 := c_{17}
$$
if $c_{16} > 0$ is small enough (recall that the support of $\vr_2$ is
contained in an interval of length $1+R$).

But, if $\vr_2(t) \ge c_{16}^2\,\ro^{-\frac{1}{2k}(d+d'-1)}$, this means that
there are at least $N$ excellent pairs $(\und{a}^1,\und{a}^{\prime
1}),\dots,(\und{a}^N,\und{a}^{\prime N})$, and for each of these pairs
$(\und{a}^\ell,\und{a}^{\prime\ell})$, an element $(\und{b}^{ij},
\und{b}^{\prime ij})$ starting with $(\und{a}^\ell,\und{a}^{\prime\ell})$, such
that $t \in J(\und{b}^{ij}, \und{b}^{\prime ij})$; renaming
$(\und{b}^{ij},\und{b}^{\prime ij}) = (\und{b}^\ell,\und{b}^{\prime\ell})$, and
setting
$$
T_{\und{b}^\ell}T_{\und{b}^{\prime\ell}}'(\und{\te},\und{\te}',s,t) =
(\und{\te}^\ell,\und{\te}^{\prime \ell},s_\ell,t_\ell),
$$
conditions (i), (ii) of 4.6 are satisfied; one has by the choice of the
$\und{b}^{ij}$, $\und{b}^{\prime ij}$:
\item{(iii)''} $(\und{\te}^\ell,\und{\te}^{\prime\ell},\tilde s) \in
\widetilde\L$ if $|\tilde s-s_\ell| \le \ro^{1/2}$;

\noindent finally, $t \in J(\und{b}^\ell,\und{b}^{\prime\ell}) \Rightarrow |t_\ell| \le 1+R$.

Hence (i), (ii), (iii)'', (iv)' are satisfied and $t \in L_{\und{0}}^{-1}(\und{\te},\und{\te}',s)$. We have proved that
$$
|L_{\und{0}}^{-1}(\und{\te},\und{\te}',s)| \ge c_{17}\,.
$$

\comment
\noindent{\bf 4.11.1}\quad Define $\ov J_1(\und{c}î, \und{c}^{\prime i})$ as the set of $t$ that belong to at least $\dfrac 12 c_3 \bar c \ro^{-\frac 12(d+d'-1)}$ intervals $J(\und{b}^{ij}, \und{b}^{\prime ij})$ starting with $(\und{c}î, \und{c}^{\prime i})$, where $\bar c$ is a small constant to be choosed later; the define $\ov J_2(\und{a},\und{a}')$ as the set of $t$ that belong to at least $\bar c\ro^{(\frac{1}{2k}-\frac 12)(d+d'-1)}$ sets $\ov J_1(\und{c}î, \und{c}^{\prime i})$, with $(\und{c}î, \und{c}^{\prime i})$ starting with $(\und{a}, \und{a}')$. (Take $k$ is so large that $\bigg(\dfrac 12 - \dfrac{1}{2k}\bigg)(d+d'-1) > \dfrac 12 d^*$).

Finally, $\ov\vr_2$ is defined as the sum over excellent pairs $(\und{a}, \und{a}')$ of the characteristic functions of $\ov J_2(\und{a}, \und{a}')$.

Arguments analogous as the previous give $m(\{\ov\vr_2 \ge \bar c\ro^{-\frac{1}{2K}(d+d'-1)}\}) \ge \ov{c_{17}}$\,, and it gives the desired estimate $|\ov L_{\und{0}}^{-1}(\und{\te}, \und{\te}',s)| \ge \ov c_{17}$\,, and we can use the arguments in the end of the section to show $m(\ov L_{\und{\om}}^{-1}(\und{\te}, \und{\te}',s)) \ge \ov{c_{17}}$\,.
\endcomment

\vglue .2in

\noindent{\bf 4.12}\quad We now consider the random perturbation
$g^{\und{\om}}$ instead of $g$. By definition, we have to work with
$g^{\und{\om}^*}$.

By 4.7, doing this, the endpoints of each $J(\und{a},\und{a}')$ are moved by no
more than $c_{18}c_5\,\ro^{1-\frac{1}{2k}}$. So if $\ro$ is small enough, and
$(\und{a},\und{a}')$ is good, there will be (for $g^{\und{\om}^*}$) no more
than $c_{16}^{-1}\,\ro^{-\frac{1}{2k}(d+d'-1)}$ intervals $J(\tilde a, \tilde
a')$ whose centers are distant from the center of $J(a,a')$ by less than
$\dfrac 74 c_{19}\,\ro^{\frac{1}{2k}}$ [instead of $2c_{19}\,\ro^{1/2k}$ in the
unperturbed situation].

When we consider the pairs $(\und{c}^i,\und{c}^{\prime i})$,
$(\und{d}^{ij},\und{d}^{\prime ij})$, $(\und{b}^{ij}, \und{b}^{\prime ij})$ of
the beginning of 4.8, the conditions
$\und{\te}^i \in \Sigma_{nr}^-$\,, \,\, $\und{\te}^{ij} \in \Sigma_{nr}^-$ are
independent of $\und{\om}$. If we write
$$
T_{\und{c}_i}^{\und{\om}^*}T_{\und{c}_i'}'(\und{\te},\und{\te}',s) =
(\und{\te}^i,\und{\te}^{\prime i},s_i(\und{\om}^*)),
$$
we have $|s_i(\und{\om^*})-s_i| \le c_5c_{18}'\,\ro^{1-\frac{1}{2k}}$ by Lemma
4.7 (ii); writing
$$
\align
&T_{\und{d}_{ij}}^{\und{\om}^*} T_{\und{d}_{ij}'}'(\und{\te}^i,\und{\te}^{\prime
i},s_i) = (\und{\te}^{ij}, \und{\te}^{\prime ij}, s_{ij}^0(\und{\om}^*)),\\
&T_{\und{d}_{ij}}^{\und{\om}^*} T_{\und{d}_{ij}'}'(\und{\te}^i,\und{\te}^{\prime
i},s_i(\und{\om}^*) = (\und{\te}^{ij}, \und{\te}^{\prime ij},
s_{ij}(\und{\om}^*)),
\endalign
$$
we have
$$
\align
&|s_{ij}(\und{\om}^*) - s_{ij}^0(\und{\om}^*)| \le c_5c_{18}''\,\ro^{1-
\frac{1}{2k}},\\
&|s_{ij}^0(\und{\om}^*) - s_{ij}| \le c_5c_{18}'\,\ro^{1-\frac{1}{2k}},
\endalign
$$
and thus
$$
|s_{ij} - s_{ij}(\und{\om}^*)| \le c_5(c_{18}' + c_{18}'')\ro^{1-\frac{1}{2k}}.
$$
This will guarantee condition (iii)' of 4.6 later.

Keeping the notion of excellent pairs to be the same, we consider for the
perturbed map $g^{\und{\om}^*}$ and excellent pair $(\und{a},\und{a}')$ the
subsets $J_1^{\und{\om}^*}(\und{a},\und{a}')$,
$J_2^{\und{\om}^*}(\und{a},\und{a}')$ of $J^{\und{\om}^*}(\und{a},\und{a}')$.

By the estimates of Lemma 4.7 and the Lemma 1 of 4.10, we will still have
$$
|J_1^{\und{\om}^*}(\und{a},\und{a}')| \ge c_{22}^{\prime -
1}\,c_{16}\,\ro^{\frac{1}{2k}}.
$$
For the same reasons, we will get 
$$
|J_1^{\und{\om}^*}(\und{c}^i,\und{c}^{'i})| \ge c_{23}^{\prime -
1}|J^{\und{\om}^*}(\und{c}^i,\und{c}^{\prime i})|
$$
and this allows to prove in exactly the same way that
$$
m\big(\big\{\vr_2^{\und{\om}^*} \ge c_{16}^2\,\ro^{-\frac{1}{2k}(d+d'-
1)}\big\}\big) \ge c_{17}'\,.
$$
The proof of
$$
\vr_2^{\und{\om}^*}(t) \ge c_{16}^2\,\ro^{-\frac{1}{2k}(d+d'-1)} \Rightarrow t
\in L_{\und{\om}}^{-1}(\und{\te},\und{\te}',s)
$$
is the same, except that we have
$$
|s_\ell(\und{\om}^*) - s_\ell| \le c_5(c_{18}' + c_{18}'')\ro^{1-\frac{1}{2k}},
$$
which gives (iii)' instead of (iii)''.

\vglue .2in

\noindent{\bf 4.13}\quad If we want to prove the reinforcement of theorem 2.4 given in 2.5, one has to modify the definition of $L_{\und{\om}}^0(\und{\te},\und{\te}',s)$ (for $(\und{\te}, \und{\te}',s) \in \widetilde\L$, $\und{\om} \in \Om$) as follows.

Let $\tilde d < d + d'-1$; choose $\tilde d < d^* < d + d'-1$, and $k$ large enough to have
$$
\bigg(1-\frac 1k\bigg)(d+d'-1) > d^*.
$$

Let $M = [\ro^{-d^*}]$. We define $\ov L_{\und{\om}}^0(\und{\te},\und{\te}',s)$ to be set of points $(\und{\te},\und{\te}',s,t)$ in the fiber for which there exist pairs $(\und{b}_m^\ell, \und{b}_m^{'\ell})$, $1 \le \ell \le N$,\,\, $1 \le m \le M$ such that conditions (ii), (iii), (iv) of \S4.6 are satisfied for all $\ell$, $m$ and moreover

\item{(i')} for any $1 \le m \le M$, the words $\und{b}_m^1,\dots,\und{b}_m^N$ are independent.

We also ask that 
\item{(v)}\qquad $I(\und{b}_m^\ell) \cap I(b_{\und{m}'}^{\ell '}) = \phi$ if $(\ell,m) \ne (\ell', m')$.

Replacing (iii), (iv) by (ii)', (iv)' we also define $\ov L_{\und{\om}}^{-1}(\und{\te},\und{\te}',s)$.

We now claim that proposition 4.6 still holds:
$$
|\ov L_{\und{\om}}^{-1}(\und{\te},\und{\te}',s)| > \bar c_{17}\,.
$$
To prove this, we proceed as above: for an excellent pair $(\und{a}, \und{a}')$ and $(\und{c}^i, \und{c}^{'i})$ starting with $(\und{a}, \und{a}')$, we now define $\ov J_1(\und{c}^i, \und{c}^{'i})$ as the set of $t$ which belong to at least $\ro^{-\frac 12d^*}$ intervals $J(\und{b}^{ ij}, \und{b}^{'ij})$ (with $(\und{b}^{ij}, \und{b}^{'ij})$ starting with $(\und{c}^i, \und{c}^{'i})$). As $\dfrac 12\,d^*  < \dfrac 12\,(d+d'-1)$, one easily sees that we still have
$$
|\ov J_1(\und{c}^i, \und{c}^{'i})| \ge \bar{c}_{23}^{-1}|J(\und{c}^i, \und{c}^{'i})|.
$$

Next we define $\ov J_2(\und{a}, \und{a}')$ as the set of $t$ which belong to at least $\ro^{-1/2\,d^*}$ sets $\ov J_1(\und{c}^i, \und{c}^{'i})$ (with $(\und{c}^i, \und{c}^{'i})$ starting with $(\und{a}, \und{a}')$). Because $\dfrac 12\,d^* < \bigg(\dfrac 12 - \dfrac{1}{2k}\bigg)(d+d'-1)$, we will still have
$$
|\ov J_2(\und{a}, \und{a}')| \ge \bar{c}_{23}^{-1}\,\ro^{1/2k}.
$$
If $\bar\vr_2$ is the sum, over excellent pairs, of the characteristic functions of the $\ov J_2(\und{a},\und{a}')$, the same argument than for $\vr_2$ shows that
$$
m(\{\bar\vr_2 \ge N\}) \ge \bar{c}_{17}\,.
$$
If $\bar\vr_2(t) \ge N$, it is now easy to construct the $(\und{b}_m^\ell, \und{b}_m^{'\ell})$ as required: the index $\ell$ refers to the excellent pairs such that $t \in \ov J_2(\und{a}, \und{a}')$, and the index $M \le \ro^{-\frac 12\,d^*}\cdot\ro^{-\frac 12\,d^*}$ for the possibilities for fixed $(\und{a}, \und{a}')$.

\vglue .5in

\noindent{\bf \S5.\quad The family of random perturbations.}

\vglue .2in

\noindent{\bf 5.1}\quad In this section, we will prove the estimates of lemma \S4.7 and the proposition \S3.8. We begin with lemma \S4.7, part (i).

We are given $\und{\te} \in \Sigma^-$ and $\und{a} \in \Sigma^+$, with $a_0 =\te_0$\,. Let $\und{\om} \in \Om$; we want to show that
$$
|k^{\und{\te},\und{\om}}(h^{\und{\om}}(\und{a})) - k^{\und{\te}}(h(\und{a}))|
\le c_{18}\,c_5\,\ro^{1-\frac{1}{2k}}.
$$
Let $\hat a = (a_0,\dots,a_m)$ be an initial part of $\und{a}$ which belongs to
$\Sigma(\ro^{1-1/2k})$. The sizes of $I^{\und{\te}}(\hat a)$ and
$I^{\und{\te},\und{\om}}(\hat a)$ have order $\ro^{1-1/2k}$, and we have to show that these two intervals are distant by no more than $c_5c_{18}\,\ro^{1-1/2k}$.

To simplify notation, we just write $\tilde f_{aa'}$ for the inverse branches associated to $g^{\und{\om}}$.

Because the $C^0$-distance between $f_{aa'}$ and $\tilde f_{aa'}$ is
$c_5\,\ro^{1+1/2k}$, and these maps are contractions, the endpoints of
$I^{\und{\om}}(\hat a)$ are distant from those of $I(\hat a)$ by no more than a multiple of $c_5\,\ro^{1+1/2k}$. So it is sufficient to see that the $C^0$-distance between $k^{\und{\te}}$ and $k^{\und{\te},\und{\om}} := \tilde k^{\und{\te}}$ is no more than $c_{18}c_5\, \ro^{1-1/2k}$.

\vglue .2in

\noindent{\bf 5.2}\quad Let $z \in I(\te_0)$; set
$$
x_0 = \tilde x_0 = 0, \quad y_0 = \tilde y_0 = 1, \quad z_0 = \tilde z_0 = z,
$$
and define by induction
$$
\align
&x_n = f_{\te_{-n},\te_{-n+1}}(x_{n-1}), \quad y_n = f_{\te_{-n},\te_{-n+1}}(y_{n-1})\\
&z_n = f_{\te_{-n},\te_{-n+1}}(z_{n-1}), \quad \tilde x_n = \tilde f_{\te_{-n},\te_{-n+1}}(\tilde x_{n-1}),\\
&\tilde y_n = \tilde f_{\te_{-n},\te_{-n+1}}(\tilde y_{n-1}), \quad \tilde z_n = \tilde f_{\te_{-n},\te_{-n+1}}(\tilde z_{n-1}).
\endalign
$$
We will just write $f_n$\,, $\tilde f_n$ for $f_{\te_{-n},\te_{-n+1}}$, $\tilde f_{\te_{-n},\te_{-n+1}}$\,. We have
$$
\align
&k^{\und{\te}}(z) = \underset{n\to+\infty}\to{\text{Lim}}\,\frac{z_n-x_n}{y_n-x_n}\,,\\
&\tilde k^{\und{\te}}(z) = \underset{n\to+\infty}\to{\text{Lim}}\, \frac{\tilde z_n - \tilde x_n}{\tilde y_n - \tilde x_n}\,\cdot
\endalign
$$
Because $f_n$\,, $\tilde f_n$ are $C^0$-distant by $c_5\,\ro^{1+1/2k}$, we will always have
$$
\align
&|z_n-\tilde z_n| \le cc_5\,\ro^{1+1/2k},\\
&|x_n-\tilde x_n| \le cc_5\, \ro^{1+1/2k},\\
&|y_n-\tilde y_n| \le cc_5\,\ro^{1+1/2k}.
\endalign
$$
Thus, as long as $|y_n-x_n| \ge c_0^{-3}\,\ro^{-1/k}$, one has
$$
\lv \frac{z_n-x_n}{y_n-x_n} - \frac{\tilde z_n - \tilde x_n}{\tilde y_n-\tilde x_n}\rv \le cc_5\,\ro^{1-\frac{1}{2k}}.
$$
Let $n_0$ be the largest such integer; we will have $|y_n-x_n| \le c_0^{-2}\,\ro^{-1/k}$ for $n \ge n_0$\,. Therefore on an interval containing the six points $x_n$\,, $y_n$\,, $z_n$\,, $\tilde x_n$\,, $\tilde y_n$\,, $\tilde z_n$\,, we will have (cf. 3.6.3) either
$$
\tilde f_{n+1}(u) = f_{n+1}(u)
$$
or
$$
\tilde f_{n+1}(u) = f_{n+1}(u) + c_5\,\om_{n+1}\,\ro^{1+1/2k}
$$
(where $\om_{n+1}$ is a component of $\und{\om}$).

\noindent Thus we will have, for $n \ge n_0$\,:
$$
\align
&z_{n+1} - x_{n+1} = (z_n-x_n) \int_0^1 Df_{n+1}(x_n + t(z_n-x_n))\,dt,\\
&\tilde z_{n+1} - \tilde x_{n+1} = (\tilde z_n-\tilde x_n) \int_0^1
Df_{n+1}(\tilde x_n + t(\tilde z_n-\tilde x_n))\,dt,
\endalign
$$
and similarly for $y_{n+1}-x_{n+1}$\,, $\tilde y_{n+1} - \tilde x_{n+1}$\,.

Write
$$
\align
&\frac{z_{n+1} - x_{n+1}}{y_{n+1} - x_{n+1}} = \frac{z_n-x_n}{y_n-
x_n}\,\,\al_n\,,\\
&\frac{\tilde z_{n+1} - \tilde x_{n+1}}{\tilde y_{n+1} - \tilde x_{n+1}} = \frac{\tilde z_n-\tilde x_n}{\tilde y_n-\tilde x_n}\,\,\tilde\al_n\,,
\endalign
$$
with
$$
\align
\al_n &= \frac{\int_0^1 Df_{n+1}(x_n+t(z_n-x_n))dt}{\int_0^1
Df_{n+1}(x_n+t(y_n-x_n))dt}\,,\\
\tilde\al_n &= \frac{\int_0^1 Df_{n+1}(\tilde x_n+t(\tilde z_n-
\tilde x_n))dt}{\int_0^1 Df_{n+1}(\tilde x_n+t(\tilde y_n-\tilde
x_n))dt}\,\cdot
\endalign
$$
We have
$$
|\al_n-1| \le c|y_n-x_n|, \quad |\tilde\al_n-1| \le c|\tilde y_n-\tilde x_n|
$$
and also
$$
|\al_n-\tilde\al_n| \le cc_5\,\ro^{1+1/2k}.
$$
We obtain, setting
$$
\frac{\tilde z_{n+1} - \tilde x_{n+1}}{\tilde y_{n+1} - \tilde x_{n+1}} -\frac{z_{n+1} - x_{n+1}}{y_{n+1} - x_{n+1}} = w_{n+1}
$$
that
$$
\align
|w_{n+1}| &\le |w_n|(1+c|y_n-x_n|)\\
&+ c\text{ Min }(c_5\,\ro^{1+1/2k}, |x_n-y_n|+|\tilde x_n-\tilde y_n|),
\endalign
$$
starting with$$
|w_{n_0}| \le cc_5\,\ro^{1-1/2k}.
$$
It is thus clear that $|w_n| \le cc_5\,\ro^{1-1/2k}$ for all $n \ge n_0$\,, which gives part (i) of Lemma \S4.7.

\vglue .2in

\noindent{\bf 5.3}\quad We now turn to the second part of Lemma \S4.7.

Let $\und{\te} \in \Sigma^-$, and $\und{a} = a_0\dots a_m$ a word in $\Sigma$
with $a_0 = \te_0$ and $|I(\und{a})| > c_0^{-1}\,\ro$. We define
$$
\alignedat{2}
&u_{-m} = \tilde u_{-m} = 0, &&\quad v_{-m} = \tilde v_{-m} = 1\\
&u_{-j} = f_{a_ja_{j+1}}(u_{-j-1}), &&\quad v_{-j} = f_{a_ja_{j+1}}(v_{-j-1})\\
&\tilde u_{-j} = \tilde f_{a_ja_{j+1}}(\tilde u_{-j-1}), &&\quad \tilde v_{-j}
= \tilde f_{a_ja_{j+1}}(\tilde v_{-j-1}),
\endalignedat
$$
for $0 \le j < m$. Then $u_0$\,, $v_0$ (resp. $\tilde u_0$\,, $\tilde v_0$) are
the endpoints of $I(\und{a})$ (resp. $I^{\und{\om}}(\und{a}$)). For $n \ge 0$, we define $(x_n)\_{n\ge0}$\,, $(y_n)\_{n\ge0}$\,, $(u_n)\_{n\ge0}$\,, $(v_n)\_{n\ge0}$\,, $(\tilde
x_n)\_{n\ge0}$\,, $(\tilde y_n)\_{n\ge0}$\,, $(\tilde u_n)\_{n\ge0}$\,,
$(\tilde v_n)\_{n\ge0}$ as in 5.2. One has
$$
\frac{|I^{\und{\te},\und{\om}}(\und{a})|}{|I^{\und{\te}}(\und{a})|} =
\underset{n\to+\infty}\to{\text{Lim}} \frac{|\tilde v_n-\tilde u_n|}{|v_n-u_n|}
\frac{|y_n-x_n|}{|\tilde y_n-\tilde x_n|}\,\cdot
$$
As above, we have
$$
\align
&\text{Max} (|v_n-\tilde v_n|, |u_n-\tilde u_n|) \le cc_5\,\ro^{1+1/2k},\,\,\,
\forall\, n \ge -m\\
&\text{Max}(|x_n-\tilde x_n|, |y_n-\tilde y_n|) \le cc_5\,\ro^{1+1/2k},\,\,\,
\forall\, n \ge 0.
\endalign
$$
Let $n_1$ be the largest integer for which $|v_n-u_n| \ge c_0^{-3}\,\ro^{-
1/k}$; we have
$$
\lv \text{ Log } \frac{|\tilde v_{n_1}-\tilde u_{n_1}|}{|v_{n_1} - u_{n_1}|}\rv
\le cc_5\,\ro^{1-1/2k}
$$
and, for $n \ge n_1$
$$
\frac{\tilde v_{n+1} - \tilde u_{n+1}}{v_{n+1} - u_{n+1}} = \frac{\tilde v_n -
\tilde u_n}{v_n - u_n}\,\,\la_n\,,
$$
with
$$
\la_n = \frac{\int_0^1 Df_{n+1}(\tilde u_n +t(\tilde v_n-\tilde
u_n))dt}{\int_0^1 Df_{n+1}(u_n+t(v_n-u_n))dt}
$$
so that
$$
|\la_n-1| \le cc_5\,\ro^{1+1/2k}.
$$
This means that for $n_2 = [\ro^{-1/k}] \gg n_1$\,, we will still have
$$
\lv \text{ Log } \frac{|\tilde v_{n_2}-\tilde u_{n_2}|}{|v_{n_2} - u_{n_2}|}\rv
\le cc_5\,\ro^{1-1/2k}.
$$
We can show in the same way that
$$
\lv \text{ Log } \frac{|\tilde y_{n_2}-\tilde x_{n_2}|}{|y_{n_2} - x_{n_2}|}\rv
\le cc_5\,\ro^{1-1/2k}
$$
For $n \ge n_2$\,, we write as in 5.2
$$
\frac{|v_{n+1} - u_{n+1}|}{|y_{n+1} - x_{n+1}|} = \frac{|v_n-u_n|}{|y_n-x_n|}\,\al_n
$$
with $|\al_n-1| \le c|y_n-x_n|$; this allows to get
$$
\underset{n\to+\infty}\to{\text{Lim}}\,\frac{|v_n-u_n|}{|y_n-x_n|} =
\frac{|v_{n_2}-u_{n_2}|}{|y_{n_2}-x_{n_2}|}\,\al,
$$
with $|\al-1| \le c|y_{n_2}-x_{n_2}|$ (because $|y_n-x_n|$ is decreasing geometrically). Similarly
$$
\underset{n\to+\infty}\to{\text{Lim}}\,\frac{|\tilde v_n-\tilde u_n|}{|\tilde y_n-\tilde x_n|} = \frac{|\tilde v_{n_2}-\tilde u_{n_2}|}{|\tilde y_{n_2}- \tilde x_{n_2}|}\,\tilde\al,
$$
with $|\tilde\al-1| \le c|\tilde y_{n_2}-\tilde x_{n_2}|$. As we have
$$
\text{Max}(|y_{n_2} - x_{n_2}|, |\tilde y_{n_2} - \tilde x_{n_2}|) \ll \ro^{1-
1/2k}.
$$
we obtain the required estimate.

\vglue .2in

\noindent{\bf 5.4}\quad In the rest of this section, we will prove the
Proposition \S3.8.

We first recall the setting.

We are given $u = (\und{\te},\und{\te}',s,t) \in \widetilde\L \times \re$.

The set $\Om_1(u)$ is the set of parameters $\und{\om} \in \Om$, such that there exists $\tilde u = (\und{\widetilde\te}, \und{\widetilde\te}',\tilde s,
\tilde t) \in \widetilde\L \times \re$ with
$$
d(\und{\te},\und{\widetilde\te}) \le 2\ro^{5/2}, \quad d(\und{\te}',
\und{\widetilde\te}') \le 2\ro^{5/2}, \quad |s-\tilde s| < \ro, \quad |t-\tilde t| < \ro
$$
and $\tilde t \in L_{\und{\om}}^0(\widetilde\te, \widetilde\te', \tilde s)$.

\noindent The set $\Om_0(u)$ is the set of parameters $\und{\om} \in \Om$ for which there exists $\und{b} \in \Sigma(\ro)$, $\und{b}' \in
\Sigma'(\ro)$ (with $b_0 = \te_0$\,, $b_0' = \te_0'$) such that the image
$$
T_{\und{b}}^{\und{\om}} T_{\und{b}'}'(u) = \hat u = (\und{\widehat\te},
\und{\widehat\te}', \hat s, \hat t)
$$
satisfies
$$
\align
(\widehat\te, \widehat\te', \hat s') &\in \L \text{ for } |\hat s-\hat s'| <
\frac 12 \ro^{1/2}\\
&\hat t \in L_{\und{\om}}^0(\widehat\te, \widehat\te', \hat s).
\endalign
$$
We have to prove that, provided $c_5$ is large enough
$$
\bp(\Om^1(u) - \Om^0(u)) \le \exp(-c_7\,\ro^{-\frac{1}{2k}(d+d'-1)})
$$
where $\bp$ is normalized Lebesgue measure on $\Om$.

\vglue .2in

\noindent{\bf 5.5}\quad Recall the decomposition of 4.5:
$$
\align
&\Om = [-1,+1]^{\Sigma_1(\und{\te})} \times [-1,+1]^{\Sigma_1-
\Sigma_1(\und{\te})}\\
&\und{\om} = (\und{\om}', \und{\om}'')
\endalign
$$
which only depends on $\und{\te}$ through an endword in $\Sigma(\ro^{1/2k})$.
We have set $\und{\om}^* = (\und{0},\und{\om}'')$ in \S4.5, and
$L_{\und{\om}}^0(\und{\te},\und{\te}',s)$ actually only depend on $\und{\om}^*$
(or $\und{\om}''$), {\it not\/} on $\und{\om}'$.

Therefore, the property $\und{\om} \in \Om^1(u)$ depends only on $\und{\om}''$ (one has $\Sigma_1(\und{\widetilde\te}) = \Sigma_1(\und{\te})$ as $d(\und{\te},
\und{\widetilde\te}) \le 2\ro^{5/2})$.

We will fix $\und{\om}'' \in \Om'' = [-1,+1]^{\Sigma_1-\Sigma_1(\und{\te})}$.
 Then either $\und{\om}^* = (\und{0},\und{\om}'') \notin \Om^1(u)$, and then $([-1,+1]^{\Sigma_1(\und{\te})} \times \{\und{\om}''\}) \cap \Om^1(u)$ is empty, or $\und{\om}^* \in \Om^1(u)$. In this last case, we will prove that
$$
\bp_{\Om'}(\Om' - \Om^{\prime 0}(u)) \le \exp(-c_7\,\ro^{-\frac{1}{2k}(d+d'-
1)}),
$$
where
$$
\align
&\Om' = [-1,+1]^{\Sigma_1(\und{\te})},\\
&\Om^{\prime 0}(u) = \{\om' \in \Om', (\om', \om'') \in \Om^0(u)\},
\endalign
$$
and $\bp_{\Om'}$ is Lebesgue measure normalized on $\Om'$. Proposition \S3.8
will then follow by Fubini's theorem.

\vglue .2in

\noindent{\bf 5.6}\quad Now on, $\und{\om}''$ is fixed, with $\und{\om}^* \in
\Om^1(\und{u})$. This means that there exists $\tilde u = (\widetilde\te,
\widetilde\te', \tilde s, \tilde t)$ in $\widetilde\L\times\re$ such that
$$
\align
&d(\und{\te},\und{\widetilde\te}) \le 2\ro^{5/2}, \quad
d(\und{\te}',\und{\widetilde\te}') \le 2\ro^{5/2},\quad |s-\tilde s| < \ro, \quad |t-\tilde t| < \ro\\
&\text{and } \tilde t \in L_{\und{\om}^*}^0(\widetilde\te,
\widetilde\te',\tilde s).
\endalign
$$
By definition of $L_{\und{\om}^*}^0$\,, there exist pairs
$(\und{b}^1,\und{b}^{\prime 1}),\dots,(\und{b}^N,\und{b}^{\prime N})$\quad (with $b_0^i = \widetilde\te_0 = \te_0$\,, $b_0^{\prime i} = \widetilde\te_0' = \te_0'$), with $N = [c_{16}^2\,\ro^{-\frac{1}{2k}(d+d'-1)}]$, such that, writing
$$
\align
T_{\und{b}^i}^{\und{\om}^*} T_{\und{b}^{\prime i}}'(\und{\widetilde\te},
\und{\widetilde\te}', \tilde s, \tilde t) &=: \tilde u_i\\
&=: (\und{\widetilde\te}^i, \und{\widetilde\te}^{\prime i}, \tilde s_i, \tilde
t_i)
\endalign
$$
we have

\item{(i)} $\und{b}^1,\dots,\und{b}^N$ are pairwise independent;
\item{(ii)} $\und{\widetilde\te}^i \in \Sigma_{nr}^-$ for $1 \le i \le N$;
\item{(iii)} $(\und{\widetilde\te}^i,\und{\widetilde\te}^{\prime i}, \tilde s_i') \in \L^-$ for $1 \le i \le N$,\quad $|\tilde s_i-\tilde s_i'| \le \dfrac 23 \ro^{1/2}$;
\item{(iv)} $|\tilde t_i| \le 2(1+R)$ for $1 \le i \le N$.

\noindent For $1 \le i \le N$, consider
$$
u_i = (\und{\te}^i, \und{\te}^{\prime i}, s_i, t_i) :=
T_{\und{b}^i}^{\und{\om}^*} T_{\und{b}^{\prime i}}'(\und{\te}, \und{\te}', s,t),
$$
and more generally, for $\und{\om}' \in \Om'$,\,\, $\und{\om} =
(\und{\om}',\om'')$:
$$
u_i(\und{\om}') = (\und{\te}^i, \und{\te}^{\prime i}, s_i(\und{\om}'),
t_i(\und{\om}')) := T_{\und{b}^i}^{\und{\om}} T_{\und{b}^{\prime i}}'(\und{\te},\und{\te}', s,t).
$$

\vglue .2in

\noindent{\bf Lemma}: We have, for $1 \le i \le N$,\,\, $\und{\te}^i \in
\Sigma_{nr}^-$ and, $\forall\, \und{\om}' \in \Om'$, $(\und{\te}^i,
\und{\te}^{\prime i}, s_i') \in \widetilde\L$ if $|s_i' - s_i(\om')| < \dfrac12
\ro^{1/2}$.

\vglue .1in

\noindent{\bf Proof}: From $d(\und{\te},\und{\widetilde\te}) \le 2\ro^{5/2}$,
it follows that $d(\und{\te}^i, \und{\widetilde\te}^i) \le c\,\ro^{7/2}$; but
$\und{\widetilde\te}^i \in \Sigma_{nr}^-$ and this set is determined by a
condition on an endword in $\Sigma(\ro^3)$. Therefore $\und{\te}^i \in
\Sigma_{nr}^-$\,. Moreover, $\widetilde\L$ has the same fiber over
$(\und{\widetilde\te}^i, \und{\widetilde\te}^{\prime i})$. By proposition
\S1.3, $k^{\und{\te}^i}$ and $k^{\und{\widetilde\te}^i}$ are at a $C^1$
distance of order at most $\ro^{5/2}$, and similarly for $k^{\prime
\und{\te}^{\prime i}}$ and $k^{\prime\und{\widetilde\te}^{\prime i}}$.
Therefore
$$
|\tilde s_i-s_i| \le c|\tilde s-s| + c\,\ro^{5/2} \le c\,\ro.
$$
Also, by Lemma 4.6, part (ii):
$$
|\tilde s_i(\und{\om}')-\tilde s_i| \le cc_5\,c_{18}\,\ro^{1-1/2k} \le
\frac{1}{10} \ro^{1/2}
$$
if $\ro$ is small enough.

\noindent By property (iii) above, we therefore will have
$$
(\und{\te}^i, \und{\te}^{\prime i},s_i') \in \widetilde\L \text{ if } |s_i' -
\tilde s_i(\und{\om}')| < \frac 12 \ro^{1/2}. \qquad\qed
$$

\vglue .2in

\noindent{\bf 5.7}\quad Let $\und{\om}' \in \Om'$; in view of the lemma above,
if there exists $1 \le i \le N$ with
$$
t_i(\und{\om}') \in L_{\und{\om}}^0(\und{\te}^i, \und{\te}^{\prime i},
s_i(\und{\om}'))
$$
then $\und{\om}' \in \Om^{\prime 0}(u)$.

\noindent Let $\und{a} \in \Sigma_2^-$\,, $\und{a} = (\te_{-r},\dots,\te_0)$ be
the endword of $\und\te$ in $\Sigma_2^-(\subset \Sigma(\ro^{1/2k}))$. For $1
\le i \le N$, let
$$
\und{a}^i = (\te_{-r},\dots,\te_0 = b_0^i,b_1^i,\dots,b_{r_i}^i)
$$
be the element in $\Sigma_1(\und{\te})$ obtained by concatenation of $\und{a}$
and the beginning of $\und{b}^i$\,. Because $\und{b}^1,\dots,\und{b}^N$ are
pairwise independent, the elements $\und{a}^1,\dots,\und{a}^N$ of
$\Sigma_1(\und{\te})$ are distinct.

Let $\widetilde\Sigma_1(\und{\te}) = \Sigma_1(\und{\te}) -
\{\und{a}^1,\dots,\und{a}^N\}$; we will just denote by $\om_i$ \,\,$(1 \le i \le N)$ the coordinate of $\und{\om}'$ corresponding to $\und{a}^i$. We thus have
$$
\align
&\Om' = [-1,+1]^N \times \widetilde\Om',\quad \widetilde\Om' = [-
1,+1]^{\widetilde\Sigma_1(\te)}\\
&\und{\om}' = ((\om_1,\dots,\om_N), \und{\tilde\om}').
\endalign
$$
We will prove that for any $\und{\tilde\om}' \in \widetilde\Om'$, the set of
$(\om_1,\dots,\om_N)$ for which there exists $1 \le i \le N$ with
$t_i(\und{\om}') \in L_{\und{\om}}^0(\und{\te}^i, \und{\te}^{\prime i},
s_i(\und{\te}'))$ fills $[-1,+1]^N$ except for a set of normalized Lebesgue measure $< \exp(-c_7\,\ro^{-\frac{1}{2k}(d+d'-1)})$ (provided $c_5$ is large enough). By Fubini's theorem, this will imply what we want.

\vglue .2in

\noindent{\bf 5.8}\quad For $\und{\tilde\om}' \in \widetilde\Om'$, set
$\und{\hat\om} = ((\und{0},\und{\tilde\om}'),\und{\om}'')$.

For $1 \le i \le N$, define
$$
L^i := L_{\und{\hat\om}}^{-1}(\und{\te}^i, \und{\te}^{\prime i},s_i(\hat\om))
\quad\text{(cf. \S4.5).}
$$
Choose $\und{\ov\te} \in \Sigma^-$ with $d(\und{\ov\te},\und{\te}) <
\ro^{5/2}$, such that the endword $\und{a} \in \Sigma_2^-$ of $\und{\te}$ (and $\und{\ov\te}$) does not appear elsewhere in $\und{\ov\te}$. For $\und{\om}' = (\om_1,\dots,\om_N, \tilde\om')$, $\und{\om} = (\und{\om}', \und{\om}'')$, consider
$$
T_{\und{b}^i}^{\und{\om}} T_{\und{b}^{\prime i}}'(\und{\ov\te},\und{\te}',s,t) =
(\und{\ov\te}^i, \und{\te}^{\prime i}, \bar s_i(\und{\om}'), \bar
t_i(\und{\om}')).
$$

\vglue .2in

\noindent{\bf Lemma 1}: $\bar t_i(\und{\om}')$ depends only (for fixed
$\und{\om}''$) on $\om_i$\,, not on $\tilde\om'$ and $\om_j$\,, $j\ne i$. Moreover, there exists $c_7' > 0$ such that
$$
m\{\om_i, \bar t_i(\om_i) \in L^i\} \ge c_7'\,.
$$

\vglue .2in

\noindent{\bf Lemma 2}: Let $\und{\om}' = (\om_1,\dots,\om_N, \und{\tilde\om}')
\in \Om'$ and $1 \le i \le N$. If $\bar t_i(\om_i) \in L^i$, then
$t_i(\und{\om}') \in L_{\und{\om}}^0(\und{\te}^i, \und{\te}^{\prime i}, s_i(\und{\om}'))$, and thus $\und{\om}' \in \Om_0'(u)$.

Clearly the two lemmas imply Proposition \S3.8, if we recall that $N =
[c_{16}^2\,\ro^{-1/2k(d+d'-1)}]$.

\vglue .2in

\noindent{\bf 5.9}\quad {\bf Proof of Lemma 1}.

Let $1 \le i \le N$. We want to understand how the endpoints of
$K^{\und{\ov\te},\und{\om}}(\und{b}^i)$ depend on $\und{\om}'$.

Consider first the sequence defined by
$$
x_0 = x_0(\und{\om}') = 0, \quad x_n(\und{\om}') = f_{\ov\te_{-n}\ov\te_{- n+1}}^{\und{\om}}(x_{n-1}(\und{\om}')) \text{ for } n > 0.
$$

{\bf Claim}: for $n \ge 0$,\,\,\, $x_n$ does not depend on $\und{\om}'$. 

By \S3.6.3, it is sufficient to prove that for $n > 0$,\,\, $x_n(\und{\om}')$ does not belong to any $\widehat I(\und{\tilde a})$, $\und{\tilde a} \in \Sigma_1(\und{\te})$.

If $|I(\ov\te_{-n},\dots,\ov\te_0)| \le \ro^{\frac{3}{5k}}$ this is true because elements of $\Sigma_1(\und{\te})$ start with $\und{a}$, and there is only one occurence of $\und{a}$ in $\und{\ov\te}$:\,\, $\und{a} = (\ov\te_{-r},\dots,\ov\te_0) \in \Sigma(\ro^{1/2k})$.

If $|I(\ov\te_{-n},\dots,\ov\te_0)| > \ro^{3/5k}$, \,\, $x_n(\und{\om}')$ cannot in fact belong to any $\widehat I(\und{\tilde a})$, $\tilde a \in \Sigma_1$\,, at least if $\ro$ is small enough; indeed, the endpoints of $K(\te_0)$ are preperiodic under $g$, so $\tilde a$ would end with a preperiodic word (independent of $\ro$) of approximate size $\ro^{2/5k}$, which precludes $\tilde a \in \Sigma_1$\,. \qquad\qed

In the same way, one proves that the sequence
$$
y_0 = y_0(\und{\om}') = 1,\qquad y_n(\und{\om}') = f_{\ov\te_{-n},\ov\te_{-
n+1}}^{\und{\om}} (y_{n-1},(\und{\om}')),
$$
is independent of $\und{\om}'$.

Consider now, with $\und{b}^i = (b_0^i,\dots,b_m^i)$, the sequence
$$
\align
&u_{-m} = u_{-m}(\und{\om}') = 0,\\
&u_{-j}(\und{\om}') = f_{b_j^i, b_{j+1}^i}^{\und{\om}}(u_{-j-
1}(\und{\om}')),\,\, 0 \le j < m\\
&u_n(\und{\om}') = f_{\ov\te_{1-n},\ov\te_{-n+1}}^{\und{\om}}(u_{n-
1}(\und{\om}')),\,\,\, n > 0.
\endalign
$$
For the same reasons than above (using that $\und{a}$ does not appear twice in $\und{\ov\te}^i$), we have that $u_n(\und{\om}')$ is independent of $\und{\om}'$ for $-m \le n < r$. Then we have, by \S3.6.3
$$
u_r(\und{\om}') = u_r(\und{0}) + c_5\,\om_i\,\ro^{1+1/2k}.
$$
For $n > r$,\,\, $u_n(\und{\om}')$ again cannot belong to any $\widehat
I(\tilde a)$, $\tilde a \in \Sigma_1(\und{\te})$. Setting
$$
\und{\overset{=}\to{\te}} = \sigma^{-r}(\und{\ov\te}) = (\dots \ov\te_{-r-1},
\ov\te_{-r})
$$
we see that the endpoint of $I^{\und{\ov\te},\und{\om}}(\und{b}^i)$ determined
by $(u_n)$ satisfies
$$
u(\und{\om}') = B\circ
k^{{\overset{=}\to{\und{\te}},\hat\om}}(u_r(0)+c_5\om_i\,\ro^{1+1/2k}),
$$
where $B$ is the affine map (with the appropriate orientation) sending
$I(\und{a})$ onto $I(\te_0)$. Neither $B$ nor
$k^{{\overset{=}\to{\und{\te}},\hat\om}}$ depend on $\und{\om}'$.

The other endpoint of $I^{\und{\ov\te},\und{\om}}(\und{b}^i)$ satisfies a similar formula.

This means that $\bar t_i$\,, as a function of $\und{\om}'$, only depends on $\om_i$\,. Moreover, as $k^{\overset{=}\to{\te},\hat\om}$ is bounded in the $C^1$-topology and the derivative of $B$ is of order $\ro^{-1/2k}$, we have
$$
c^{-1}c_5 < \lv \frac{\po\bar t_i}{\po\om_i}\rv < cc_5\,.
$$
On the other hand, we have $|L_i| > c_{17}$ by proposition \S4.5, and $L_i \subset [-(1+R), 1+R]$. So to end the proof of Lemma 1, we have just to control $\bar t_i(\und{0})$. Let
$$
T_{\und{b}^i}^{\und{\om}^*} T_{\und{b}^{\prime i}}'(\und{\ov\te},
\und{\te}',s,t) = (\und{\ov\te}^i, \und{\te}^{\prime i}, s_i^*, t_i^*).
$$
We have just seen that in fact $t_i^* = \bar t_i(0)$. On the other hand, we have
$$
T_{\und{b}^i}^{\und{\om}^*} T_{\und{b}^{\prime i}}'(\und{\widetilde\te},
\und{\widetilde\te}',\tilde s,\tilde t) = (\und{\widetilde\te}^i, \und{\widetilde\te}^{\prime i},
\tilde s_i, \tilde t_i),
$$
with $d(\und{\ov\te},\und{\widetilde\te}) \le 3\ro^{5/2}$,\,\,
$d(\und{\te}',\und{\widehat\te}') \le 2\ro^{5/2}$,\,\, $|s-\tilde s| \le \ro$,\,\, $|t-\tilde t| < \ro$ and
$$
|\tilde t_i| \le 2(1+R).
$$
Then $|\tilde t_i - t_i^*| \le cR$, which implies
$$
|\bar t_i(0)| \le 2(1+R)+cR;
$$
this concludes the proof of Lemma 1.

\vglue .2in

\noindent{\bf 5.10\quad Proof of Lemma 2}.

Let $\und{\om}' = (\om_1,\dots,\om_N, \und{\tilde\om}')$, $\und{\om} =
(\und{\om}', \und{\om}'')$, $\und{\hat\om} = ((0,\tilde\om'),\und{\om}'')$. Let also $1 \le i \le N$. We assume that $\bar t_i(\om_i) \in L_{\und{\hat\om}}^{-1}(\und{\te}^i,\und{\te}^{\prime i}, s_i(\hat\om))$ and we want to show that
$$
t_i(\und{\om}') \in L_{\und{\om}}^0(\und{\te}^i, \und{\te}^{\prime i},
s_i(\und{\om}')).
$$
The hypothesis means that there exist pairs $(\und{c}^1,\und{c}^{\prime
1}),\dots,(\und{c}^N, \und{c}^{\prime N})$ (with $c_0^j = \te_0^i$\,,
$c_0^{\prime j} = \te_0^{\prime i}$ in $\Sigma(\ro) \times \Sigma'(\ro)$ such
that, if 
$$
T_{\und{c}^j}^{\und{\hat\om}(i)} T_{\und{c}^{\prime j}}(\und{\te}^i,
\und{\te}^{\prime i}, s_i(\hat\om), \bar t_i(\om_i)) = (\und{\te}^{(j)},
\und{\te}^{\prime (j)}, \hat s_{(j)}, \hat t_{(j)})
$$
(where $\und{\hat\om}(i)$ is obtained from $\hat\om$ by setting the coordinates
in $\Sigma_1(\und{\te}^i)$ equal to 0), one has

\item{(i)} $\und{c}^1,\dots,\und{c}^N$ are pairwise independent;
\item{(ii)} $\und{\te}^{(j)} \in \Sigma_{nr}^-$ for $1 \le j \le N$;
\item{(iii)'} $(\und{\te}^{(j)}, \und{\te}^{\prime(j)}, \hat s_{(j)}') \in
\widetilde\L$ for $|\hat s_{(j)} - \hat s_{(j)}'| < \dfrac 34 \ro^{1/2}$;
\item{(iv)'} $|\hat t_{(j)}| \le 1+R$ for $1 \le j \le N$.

Let
$$
T_{\und{c}^j}^{\und{\om}(i)} T_{\und{c}^{\prime j}}'(\und{\te}^i,
\und{\te}^{\prime i}, s_i(\und{\om}'), t_i(\und{\om}')) = (\und{\te}^{(j)},
\und{\te}^{\prime(j)}, s_{(j)}, t_{(j)}),
$$
where $\und{\om}(i)$ is obtained from $\und{\om}$ by setting the coordinates in
$\Sigma_1(\und{\te}^i)$ equal to 0.

We will show tht for $1 \le j \le N$:

\item{(iii)}\quad $(\und{\te}^{(j)}, \und{\te}^{\prime(j)}, s_{(j)}') \in \widetilde\L$ for $|s_{(j)}' - s_{(j)}| < \dfrac 23 \ro^{1/2}$,

\item{(iv)}\quad $|t_{(j)}| \le 2(1+R)$,

\noindent and this implies Lemma 2.

It is thus sufficient to show that
$$
\align
&|s_{(j)} - \hat s_{(j)}| < \frac{1}{12} \ro^{1/2},\\
&|t_{(j)} - \hat t_{(j)}| < 1+R.
\endalign
$$
The first inequality is easy: by Lemma \S4.7, part (ii), we have
$$
|s_i(\und{\om}') - s_i(\hat\om)| \le c_5c_{18}\,\ro^{1-1/2k}
$$
and applying a second time the same estimate from this lemma (to compare
$T_{\und{c}^j}^{\und{\hat\om}(i)}$ and $T_{\und{c}^j}^{\und{\om}(i)}$) will
give
$$
|s_{(j)} - \hat s_{(j)}| < cc_5c_{18}\,\ro^{1-1/2k}
$$
which is $< \dfrac{1}{12} \ro^{1/2}$ for small enough $\ro$.

To prove the second inequality, we first compare $\bar t_i(\om_i)$ and
$t_i(\und{\om}')$. By definition, $\bar t_i(\om_i)$ is the $t$-coordinate of 
$T_{\und{b}^i}^{\und{\om}} T_{\und{b}^{\prime i}}(\und{\ov\te}, \und{\te}',
s,t)$, while $t_i(\und{\om}')$ is the $t$-coordinate of 
$T_{\und{b}^i}^{\und{\om}} T_{\und{b}^{\prime i}}(\und{\te}, \und{\te}', s,t)$.
As $d(\und{\te},\und{\ov\te}) \le \ro^{5/2}$, the $C^0$-distance between
$k^{\und{\te}}$, $k^{\und{\ov\te}}$ is at most of order $\ro^{5/2}$, and this
imply
$$
|\bar t_i(\om_i) - t_i(\und{\om}')| \le \ro^{3/2}.
$$
We next prove the

{\bf Claim}:\quad $|s_i(\und{\hat\om}) - s_i(\und{\om}')| \le
cc_5\,\ro^{1+1/2k}$

\noindent (a reforcement of Lemma \S4.7 part (ii) in this case).

\vglue .1in

\noindent{\bf Proof}: Let
$$
T_{\und{b}^i}^{\und{\om}} T_{\und{b}^{\prime i}}'(\und{\ov\te}, \und{\te}', s,t)
= (\und{\ov\te}^i, \und{\te}^{\prime i}, \bar s_i(\und{\om}'), \bar
t_i(\und{\om}')).
$$
As $d(\und{\te},\und{\ov\te}) \le \ro^{5/2}$, \,\, $k^{\und{\te},\und{\om}}$ and
$k^{\und{\ov\te},\und{\om}}$ are at a $C^1$-distance of order no more than
$\ro^{5/2}$. This implies that
$$
\align
&|s_i(\und{\hat\om}) - \bar s_i(\und{\hat\om})| \le c\,\ro^{5/2}\\
&|s_i(\und{\om}') - \bar s_i(\und{\om}')| \le c\,\ro^{5/2}.
\endalign
$$

To compare $\ov s_i(\und{\hat\om})$, $\bar s_i(\und{\om}')$, we go back to the
proof of Lemma 1. The endpoints of $I^{\und{\ov\te},\und{\om}}(\und{b}^i)$
depend, for fixed $\und{\om}''$, only on $\om_i$:
$$
\align
&u(\und{\om}') = B \circ k^{\und{\overset{=}\to{\te}},\hat\om}(u_r(0) +
c_5\om_i\,\ro^{1+1/2k}),\\
&v(\und{\om}') = B \circ k^{\und{\overset{=}\to{\te}},\hat\om}(v_r(0) +
c_5\om_i\,\ro^{1+1/2k}).
\endalign
$$
As $k^{\und{\overset{=}\to{\te}},\hat\om}$ is $C^2$-bounded, we obtain
$$
|\bar s_i(\und{\hat\om}) - \bar s_i(\und{\om}')| \le cc_5\,\ro^{1+1/2k},
$$
proving the claim. \qquad\qed

Finally, let
$$
T_{\und{c}^j}^{\und{\hat\om}(i)}\, T_{\und{c}^{\prime j}}'(\und{\te}^i,
\und{\te}^{\prime i}, s_i(\und{\om}'), t_i(\und{\om}')) = (\und{\te}^{(j)},
\und{\te}^{\prime(j)}, \tilde s_{(j)}, \tilde t_{(j)}).
$$

\vglue .2in

\noindent{\bf Lemma 3}: We have $|t_{(j)} - \tilde t_{(j)}| <
cc_5\,\ro^{1/2k}$.

The lemma is proved in the next section. Using it, we finish the proof of Lemma
2.

Let $\overset{\approx}\to{t}_{(j)}$ be the $t$-coordinate of
$$
T_{\und{c}^j}^{\und{\hat\om}(i)}\, T_{\und{c}^{\prime j}}'(\und{\te}^i,
\und{\te}^{\prime i}, s_i(\und{\hat\om}), t_i(\und{\om}')).
$$
Because $|\bar t_i(\om_i) - t_i(\und{\om}')| \le c\,\ro^{3/2}$, we have
$$
|\hat t_{(j)} - \overset{\approx}\to{t}_{(j)}| \le c\,\ro^{1/2}.
$$

Because $|s_i(\und{\om}') - s_i(\und{\hat\om})| \le cc_5\,\ro^{1+1/2k}$, we
have
$$
|\overset{\approx}\to{t}_{(j)} - \tilde t_{(j)}| \le cc_5\,\ro^{\frac{1}{2k}}.
$$

By Lemma 3, the same estimate holds for $|\tilde t_{(j)} - t_{(j)}|$. We
conclude that for $\ro$ small enough, we have indeed
$$
|t_{(j)} - \hat t_{(j)}| < 1+R,
$$
as was required. \qquad\qed

\vglue .2in

\noindent{\bf 5.11\quad Proof of Lemma 3}.

We will compare, for $\bar s \in J_R$ and $\bar t \in \re$, the $t$-coordinates
of 
$$
T_{\und{c}^j}^{\und{\om}(i)}\, T_{\und{c}^{\prime j}}'(\und{\te}^i,
\und{\te}^{\prime i}, \bar s, \bar t)
$$
and
$$
T_{\und{c}^j}^{\und{\hat\om}(i)}\, T_{\und{c}^{\prime j}}'(\und{\te}^i,
\und{\te}^{\prime i}, \bar s, \bar t).
$$
The two elements $\und{\om}(i)$, $\und{\hat\om}(i)$ have different coordinates
only at $\und{a}^1,\dots,\und{a}^N$ where those of $\und{\om}(i)$ are arbitrary
and those of $\und{\hat\om}(i)$ are 0. We observe also that, because
$\und{\te}^i \in \Sigma_{nr}^-$\,, we have
$$
\Sigma_1(\und{\te}) \cap \Sigma_1(\und{\te}^i) = \emptyset,
$$
and that the coordinates of $\und{\om}(i)$ (or $\und{\hat\om}(i))$ in
$\Sigma_1(\und{\te}^i)$ are 0.

Thus, we have to understand how the $t$-coordinate of 
$T_{\und{c}^j}^{\und{\om}(i)}\, T_{\und{c}^{\prime j}}'(\und{\te}^i,
\und{\te}^{\prime i}, \bar s, \bar t)$ depends on $(\om_1,\dots,\om_N)$, which
amounts to understand how the endpoints of
$I^{\und{\te}^i,\und{\om}(i)}(\und{c}^j)$ depend on $(\om_1,\dots,\om_N)$.

The scheme is the same as in the proof of Lemma 1. We replace $\und{\te}^i$ by
$\und{\ov\te}^i$; as now $d(\und{\te}^i, \und{\ov\te}^i) < c\,\ro^{7/2}$, this
will only move the endpoints of $I^{\und{\te}^i,\und{\om}(i)}(\und{c}^j)$ by at
most $c\,\ro^{7/2}$, and the corresponding $t$-coordinates by $c\,\ro^{5/2}$.

In the iteration process of \S5.9, again the only time where things depend on $(\om_1,\dots,\om_N)$ is when we apply $f_{\ov\te_{-r}
\ov\te_{-r+1}}^{\und{\om}(i)}$ (we use here that $\und{\te}^{(j)} \in \Sigma_{nr}^-$). We will see as in \S5.9 that the endpoints $w^{\pm}$ of $I^{\und{\ov\te}^i,\und{\om}(i)}(\und{c}^j)$ do not depend on $\om_k$\,, $k \ne i$, and the dependence on $\om_i$ is as follows
$$
w^{\pm}(\om_i) = \frac{\bar k(w_r^{\pm} + c_5\om_i\ro^{1+1/2k}) - \bar
k(u_r+c_5\om_i\ro^{1+1/2k})}{\bar k(v_r+c_5\om_i\ro^{1+1/2k})-\bar
k(u_r+c_5\om_i\ro^{1+1/2k})}\,,
$$
where $\bar k = k^{\und{\overset{=}\to{\te}},\und{\hat\om}(i)}$ does not depend on $\om_1,\dots,\om_N$\,.

\noindent Writing
$$
w^+(\om_i) = \frac{w_r^+-u_r}{v_r-u_r} \frac{\int_0^1 D\bar k(u_r+t(w_r^+-u_r)
+ c_5\om_i\ro^{1+1/2k})dt}{\int_0^1 D\bar k(u_r+t(v_r-u_r) + c_5\om_i
\ro^{1+1/2k})dt}\,,
$$
we see that
$$
\lv \frac{\po w^+(\om_i)}{\po \om_i}\rv \le cc_5\,\ro^{1+1/2k},
$$
as $\bar k$ is $C^2$-bounded. We thus have the control on the endpoints of $I^{\und{\ov\te}^i,\und{\om}(i)}(\und{c}^j)$ which guarantees the estimate of the lemma. \qquad\qed

\vglue .2in

\noindent{\bf 5.12}\quad For the theorem in 2.5, we need to prove the probability estimate in 3.11. But, with the notations of 4.13 and fixed $1 \le m \le M$, the same proof than above (with the $\ov L_\om$ instead of $L_\om$) shows that
$$
\bp(\ov\Om^1(u) - \Om_m^0(u)) \le \exp\big(-\overset{=}\to{c}_7\,\ro^{-\frac{1}{2k}(d+d'-1)}\big)
$$
where in the definition of $\Om_m^0(u)$ we require $(\und{b}, \und{b}')$ to be one the $\und{b}_m^\ell, \und{b}_{m'}^{\ell '}$\,. Thus we obtain
$$
\bp(\ov\Om^1(u) - \ov\Om^0(u)) \le M\,\exp\big(-\overset{=}\to{c}_7\,\ro^{-\frac{1}{2k}(d+d'-1)}\big)\le \exp\big(-\bar c_7\,\ro^{-\frac 12(d+d'-1)}\big).
$$

\vglue .5in

\noindent{\bf \S6.\quad Proof of the Scale recurrence Lemma}.

\vglue .2in

\noindent{\bf 6.1}\quad We will first formulate and prove a result, in a slightly more general setting, which contains the key argument in the proof of Scale recurrence Lemma.

We consider a finite alphabet $A$ and a finite set $\La$ with a map
$$
\align
&\La \mapsto A \times A\\
&\la \mapsto (\al(\la),\om(\la)).
\endalign
$$
Define
$$
\align
\La_i^j   &= \{\la, \al(\la)=i, \om(\la)=j\},\\
\La_i     &= \{\la, \al(\la)=i\},\\
\La^j     &= \{\la, \om(\la)=j\},\\
\La^{(2)} &= \{(\la,\la') \in \La^2,\, \al(\la') = \om(\la)\}.
\intertext{Set also}
&N_i^j = \#\,\La_i^j, \,\, N_i = \#\,\La_i\,, \,\, p_i^j = N_i^{-1}\,N_i^j\,.
\endalign
$$
We will assume that there exists $p^* > 0$ such that $p_i^j \ge p^*$ for all $i$, $j$.

The stochastic matrix $(p_i^j)$ has a left eigenvector $(p^i)\_{i\in A}$
satisfying
$$
\align
\sum_i p^ip_i^j &= p^j \ge p^*.\\
\sum_i p^i &= 1.
\endalign
$$
If we set
$$
\align
&p_\la^{\la'} =
\cases
0 \quad\text{if}\quad \om(\la) \ne \al(\la')\\
N_i^{-1} \,\,\text{if}\,\, \om(\la) = \al(\la') = i \in A,
\endcases\\
&p^\la = N_i^{-1}p^i, \,\,\, \la \in \La_i
\endalign
$$
then $(p_\la^{\la'})_{(\la,\la')\in\La^2}$ is again a stochastic matrix with left eigenvector $(p^\la)$. Indeed we have (with $\la' \in A$, $\al(\la') = j$)
$$
\align
\sum_\la p^\la p_\la^{\la'} &= \sum_i \sum_{\la\in\La_i^j} N_i^{-
1}\,p^i\,N_j^{-1}\\
&= \sum_i p_i^j\,p^i\,N_j^{-1} = p^j\, N_j^{-1} = p^{\la'}.
\endalign
$$
For $z = (z_\la)\_{\la\in\La} \in \co^\La$, define
$$
\align
||z||^2 &= \sum_\La p^\la|z_\la|^2\\
&= \sum_i\,p^i\, \frac{1}{N_i} \bigg(\sum_{\La_i} |z_\la|^2\bigg).
\endalign
$$
Observe that, if $(\te_\la^{\la'})$ is a family of complex numbers of modulus
$\le 1$ and we define
$$
w_\la = \sum_{\la'} p_\la^{\la'}\,\te_\la^{\la'}\,z_{\la'}
$$
we have
$$
\align
|w_\la|^2 &\le \sum_{\la'} p_\la^{\la'} \sum_{\la'} p_\la^{\la'}|z_{\la'}|^2 =
\sum_{\la'} p_\la^{\la'}|z_{\la'}|^2\\
\intertext{hence}
||w||^2 &\le ||z||^2.
\endalign
$$

\vglue .1in

\noindent{\bf 6.2}\quad In the setting above, we consider a family
$(E_\la)\_{\la\in\La}$ of bounded measurable subsets of $\re$, and a family
$a_\la^{\la'}$ of real numbers. Let also $\ro > 0$ be a small positive number.
For $\la\in\La$, $x\in\re$, $\om(\la)=i$, let
$$
n_\la(x) = N_i^{-1}\,\#\,\{\la' \in \La_i\,, [x-\ro,x+\ro] \subset E_{\la'} -
a_\la^{\la'}\}.
$$
We will assume that the following holds: there are $\Delta_0 > 0$,\,\, $0 < \kappa_0 < 1$ such that for any $\xi \in [1,\Delta_0 \ro^{-1}]$, the operator
$$
w_\la = \sum_{\la'} p_\la^{\la'}\, e^{i\xi a_\la^{\la'}}\,z_{\la'}
$$
acting on $(\co^\La,||\,\,\,||)$ has norm $\le \kappa_0$ (we know already that the norm is $\le 1$).

\vglue .2in

\noindent{\bf Proposition}: Under this hypothesis, there exists $0 < \kappa_1 < 1$, $\ve > 0$, $0 < \tau < 1$, {\it depending only on\/} $\Delta_0$, $\kappa_0$ such that, if $|E_\la| \le \ve$ for all $\la\in\La$, then
$$
\sum p^\la|E_\la^*| \le \kappa_1 \sum p^\la|E_\la|
$$
where
$$
E_\la^* = \{n_\la(x) > \tau\}.
$$

\vglue .2in

\noindent{\bf 6.3\quad Proof of proposition}.

For $\la \in \La$, $\om(\la) = i$, let:
$$
\align
&X_\la = {\bold 1}_{E_{\la}}\,,\\
&Y_\la = \frac{1}{N_i} \sum_{\la'\in\La_i} X_{\la'}(x+a_\la^{\la'}),\\
&Z_\la(x) = \frac{1}{2\ro} \int_{-\ro}^\ro Y_\la(x+t)\,dt.
\endalign
$$
We have $Z_\la \ge n_\la$ hence
$$
\sum_\la p^\la|Z_\la|\_{L^2}^2 \ge \tau^2 \sum_\la p^\la|E_\la^*|.
$$
We will show, under the hypotheses of the proposition that
$$
\sum_\la p^\la|Z_\la|\_{L^2}^2 \le \kappa_2  \sum p^\la|X_\la|\_{L^2}^2 = \kappa_2 \sum_\la p^\la|E_\la|,
$$
for some $\kappa_2 < 1$ depending only on $\kappa_0$, $\Delta_0$ and this will imply the proposition (taking $\kappa_1 = \tau = \kappa_2^{1/3}$). To prove this inequality, we will
consider the Fourier transforms of $X_\la$\,, $Z_\la$\,, $Y_\la$\,. We
normalize the Fourier transform as
$$
\widehat X_\la(\xi) = \int_{-\infty}^{+\infty} X_\la(x)e^{-i\xi x}\,dx.
$$
We then have
$$
\align
\widehat Y_\la(\xi) &= \frac{1}{N_i} \sum_{\la'\in\La_i} \widehat
X_{\la'}(\xi)\,e^{ia_\la^{\la'}\xi}\\
&= \sum_{\la'} p_\la^{\la'}\, e^{ia_\la^{\la'}\xi}\, \widehat X_{\la'}(\xi),
\endalign
$$
and
$$
\widehat Z_\la(\xi) = \frac{\sin \ro\xi}{\ro\xi} \widehat Y_\la(\xi).
$$
We estimate $\sum p^\la|\widehat Z_\la(\xi)|^2$ in various ways, depending on
$\xi$:

\vglue .1in

\noindent a) if $|\xi| \le 1$, we use
$$
\align
|\widehat Z_\la(\xi)| \le |\widehat Y_\la(\xi)| &\le \int_{-\infty}^{+\infty}
Y_\la(x)\,dx\\
&= \sum_{\la'} p_\la^{\la'} |E_{\la'}|
\endalign
$$
to get
$$
\align
\sum_\la p^\la|\widehat Z_\la(\xi)|^2 &\le \sum_\la p^\la\bigg(\sum_{\la'}
p_\la^{\la'} |E_{\la'}|^2\bigg)\\
&\le \sum_{\la'} p^{\la'}|E_{\la'}|^2\,;
\endalign
$$

\vglue .1in

\noindent b) if $1 \le |\xi| \le \Delta_0\ro^{-1}$, we use again
$$
|\widehat Z_\la(\xi)| \le |\widehat Y_\la(\xi)|
$$
and by the hypothesis of the proposition
$$
\sum p^\la|\widehat Y_\la(\xi)|^2 \le \kappa_0^2 \sum p^\la|\widehat X_\la(\xi)|^2;
$$

\vglue .1in

\noindent c) if $|\xi| \ge \Delta_0\ro^{-1}$, from 6.1, we have
$$
\sum p^\la|\widehat Y_\la(\xi)|^2 \le \sum p^\la|\widehat X_\la(\xi)|^2,
$$
but now
$$
\lv\frac{\sin \ro\xi}{\ro\xi}\rv \le \kappa_3 = \underset{u\ge\Delta_0}\to{\text{Max}}
\lv\frac{\sin u}{u}\rv < 1,
$$
hence $\sum p^\la|\widehat Z_\la(\xi)|^2 \le \kappa_3^2 \sum p^\la|\widehat X_\la(\xi)|^2$. Putting these estimates together gives
$$
\align
\int_{-\infty}^{+\infty} \sum_\la p^\la|\widehat Z_\la(\xi)|^2\,d\xi &\le [{\text{Max}}(\kappa_0,\kappa_3)]^2 \int_{-\infty}^{+\infty} \sum_\la p^\la|\widehat
X_\la(\xi)|^2\,d\xi\\
&+ 2\sum p^\la|E_\la|^2.
\endalign
$$

Now we have
$$
\align
&||Z_\la||\_{L^2}^2 = \frac{1}{2\pi} ||\widehat Z_\la||\_{L^2}^2\\
&||X_\la||\_{L^2}^2 = |E_\la| = \frac{1}{2\pi} ||\widehat X_\la||\_{L^2}^2\,,
\endalign
$$
and therefore
$$
\sum_\la p^\la|Z_\la|\_{L^2}^2 \le \kappa_2 \sum p^\la|E_\la|,
$$
with $\kappa_2 = [\text{Max}(\kappa_0,\kappa_3)]^2 + \pi^{-1}\,\ve$. If $\ve$ is small enough, we have $\kappa_2 < 1$ which concludes the proof of the proposition. \qquad\qed

\vglue .2in

\noindent{\bf Remarks}: The main point of the proposition is that $\tau$, $\ve$, $\kappa_1$ depend neither on the $E_\la$ nor on the $a_\la^{\la'}$ or the combinatorics.

\vglue .2in

\noindent{\bf 6.4}\quad In the same setting that 6.2, let $\Delta_1 > 0$ and 
$$
\hat n_\la(x) = N_i^{-1}\,\#\{\la' \in \La_i, [x-\Delta_1\ro, x+\Delta_1\ro] \cap (E_{\la'} - a_\la^{\la'}) \ne \emptyset\}.
$$
We denote by $V_\delta(E)$ the $\delta$-neighbourhood of a subset $E \subset \re$ (i.e. the set of points at distance $<\delta$ from $E$).

\vglue .2in

\noindent{\bf Corollary}: There exists $0 < \kappa_4 < 1$, $\ve_1 > 0$, $0 < \tau < 1$, depending only on $\Delta_0$\,, $\kappa_0$, and there exists $\Delta > 0$, depending only on $\Delta_0$\,, $\Delta_1$\,, $\kappa_0$ such that, if $|V_{\Delta\ro}(E_\la)| \le \ve_1$ for all $\la \in \La$, then
$$
\sum_\la p^\la|V_{\Delta\ro}(\widehat E_\la)| \le \kappa_4 \sum_\la
p^\la|V_{\Delta\ro}(E_\la)|,
$$
where $\widehat E_\la = \{\hat n_\la > \tau\}$.

\vglue .1in

\noindent{\bf Proof}: We first observe that, for any bounded subset $E$,
$$
|V_{(\Delta+\Delta_1+1)\ro}(E)| \le (1+\Delta^{-1}(\Delta_1+1))|V_{\Delta\ro}(E)|.
$$
On the other hand, if $[x-\Delta_1\ro, x+\Delta_1\ro]$ intersects $E_{\la'} - a_\la^{\la'}$\, and
$|y-x| < \Delta\ro$, then $[y-\ro, y+\ro]$ is contained in
$V_{(\Delta+\Delta_1+1)\ro}(E_{\la'})-a_\la^{\la'}$\,. 

We thus take $\Delta$ large enough to have
$$
\kappa_4 = (1+\Delta^{-1}(\Delta_1+1))\kappa_1 < 1, \quad \ve_1 = \ve(1+\Delta^{-1}(\Delta_1+1))^{-1},
$$
and apply the proposition to the $V_{(\Delta+\Delta_1+1)\ro}(E_\la)$.
\qquad\quad\qquad\qed

\vglue .2in

\noindent{\bf 6.5}\quad After these general statements, we come back to the setting of the Scale recurrence Lemma.

Recall that $\Sigma(\ro)$ is the set of words $\und{a}$ in $\Sigma$ such that
$$
c_0^{-1}\,\ro \le |I(\und{a})| \le c_0\,\ro
$$
and that we write
$$
J_R = [-R,-R^{-1}] \cup [R^{-1}, R].
$$
We will need in the proof to distinguish two cases:

A) \,\,$g$ and $g'$ are orientation preserving (after perhaps modifying the orientation of some of the $I(a)$, $I(a')$, $a \in \A$, $a'\in\A'$). Equivalently, all periodic points of $g$ {\it and\/} $g'$ have a positive eigenvalue.

B) There exists at least one periodic point of $g$ or $g'$ with negative eigenvalue.

We will do the proof in case B). Case A) is simpler and we will rapidly indicate at the end how to modify the proof.

In relation to the abstract setting of \S6.1, we set
$$
\align
&A = \AA \times \AA' \times \{-1, +1\},\\
&\La = \Sigma(\ro) \times \Sigma'(\ro) \times \{-1, +1\}.
\endalign
$$
The map $(\al,\om)\colon \La \to A \times A$ is defined as follows: for $\und{a} \in \Sigma(\ro)$, $\und{a}' \in \Sigma'(\ro)$, $u \in \{-1, +1\}$
$$
\align
&\al(\und{a}, \und{a}',u) = (a_0, a_0', uw)\\
&\om(\und{a}, \und{a}', u) = (a_n, a_{n'}'\,, u),
\endalign
$$
where $a_n$\,, $a_{n'}'$ are the last letters of $\und{a}$, $\und{a}'$ and $w=+1$ (resp. $-1$) if $f_{\und{a}}$, $f_{\und{a}'}'$ have the same (resp. opposite) orientation.

It follows from the hypothesis of case B), and the hypothesis that $\Sigma$, $\Sigma'$ are topologically mixing, that if $c_0$ has been chosen large enough we have, for all $i,j \in A$
$$
c^{-1}\,\ro^{-(d+d')} \le \#\, \La_i^j = N_i^j \le c\ro^{-(d+d')}.
$$
We are given a family $E(\und{a}, \und{a}')$ of subsets of $J_R$\,, for $\und{a} \in \Sigma(\ro)$, $\und{a}' \in \Sigma'(\ro)$. Define, for $(\und{a}, \und{a}',u) \in \La$
$$
F(\und{a},\und{a}',u) = \{x, ue^x \in E(\und{a}, \und{a}')\},
$$
which is a subset of $[-r,r]$, $r = \text{ Log } R$. To say that all $J_R - E(\und{a}, \und{a}')$ have small measure amounts to say that all $[-r,r] - F(\la)$, $\la\in\La$, have small measure:
$$
m([-r,r] - F(\la)) \le \bar c_1\,.
$$
Finally, we define the family of translations
$(a_\la^{\la'})\_{(\la,\la')\in\La^{(2)}}$ of \S6.2. To do this, we choose, for each $\und{a} \in \Sigma(\ro)$ (resp. $\und{a}' \in \Sigma'(\ro)$) an element $\und{\te} \in \Sigma^-$ (resp. $\und{\te}' \in \Sigma^{\prime -}$) ending with $\und{a}$ (resp. $\und{a}'$). Then, if $\la = (\und{a}, \und{a}',u)$, $\la' = (\und{b}, \und{b}',v)$ satisfy $\om(\la) = \al(\la')$, we set 
$$
a_\la^{\la'} = \text{ Log }|I^{\und{\te}'}(\und{b}')| - \text{ Log
}|I^{\und{\te}}(\und{b})|.
$$
We will prove later the following

\vglue .2in

\noindent{\bf Lemma}: The hypothesis of the Scale recurrence Lemma implies that the hypothesis of Proposition 6.2 holds, namely there exist $\Delta_0 > 0$, $0 < \kappa_0 < 1$ such that for $\xi \in [1,\Delta_0\ro^{-1}]$ the operator defined by
$$
w_\la = \sum p_\la^{\la'}\,e^{ia_\la^{\la'}\,\xi}\,z_{\la'}
$$
has norm $\le \kappa_0$\,.

\vglue .2in

\noindent{\bf 6.6}\quad Let $\Delta_1 > 0$, and let then $\Delta$ be given by the corollary in 6.4. For $\la \in \La$, we set
$$
E_0(\la) = [-r,r] - V_{\Delta\ro}(F(\la)).
$$

Starting from $E_0$\,, we define for $n \ge 0$ sets $E_n(\la)$ in the following way; if $E_n(\la)$ has already been defined, we define, with $c_3 > 0$ fixed sufficiently small, $\Hat{\Hat E}_n(\la)$ as the set of $x \in [-r,r]$ such that
$$
\#\{\la', \om(\la) = \al(\la'), [x+a_\la^{\la'}-\Delta_1\ro,
x+a_\la^{\la'}+\Delta_1\ro] \subset [-r,r]-E_n(\la')\}
$$
is less than $c_3\,\ro^{-(d+d')}$. We then set
$$
E_{n+1}(\la) = E_0(\la) \cup \Hat{\Hat E}_n(\la).
$$
We will see that, provided $\bar c_1$\,, $c_3$ are sufficiently small, we have

\vglue .2in

\noindent{\bf Lemma}:\quad $\sum\limits_\la p^\la|V_{\Delta\ro}
(\Hat{\Hat E}_n(\la))| \le \kappa_5
\sum\limits_\la p^\la|V_{\Delta\ro}(E_n(\la))|$, with $0 < \kappa_5 < 1$.

Before proving the lemma we finish the proof of the Scale recurrence Lemma. It is clear that for each $\la\in\La$, the sequence $(E_n(\la))\_{n\ge0}$ is increasing. On the other hand, by the lemma, we have
$$
\sum p^\la|V_{\Delta\ro}(E_{n+1}(\la))| \le \kappa_5 \sum p^\la|V_{\Delta\ro}(E_n(\la))|
+ \underset{\la}\to{\text{Max}} |V_{\Delta\ro}(E_0(\la))|.
$$
Defining $E_\infty(\la) = \underset{n\ge0}\to{\bigcup}\,E_n(\la)$, we thus obtain
$$
\sum p^\la|V_{\Delta\ro}(E_\infty(\la))| \le \frac{1}{1-\kappa_5}
\underset{\la}\to{\text{Max}} |V_{\Delta\ro}(E_0(\la))|.
$$
We observe that $V_{\Delta\ro}(E_0(\la))$ is contained in $[-r-\Delta\ro, r+\Delta\ro] - F(\la)$, hence its measure is less than $2\Delta\ro + \bar c_1$\,. Consider
$$
F_\infty(\la) = [-r, +r] - E_\infty(\la).
$$
Let $x \in F_\infty(\la)$; by definition of $\Hat{\Hat E}_n(\la) \subset E_{n+1}(\la)$, we have, for every $n \ge 0$
$$
\#\{\la', \om(\la)=\al(\la'), [x+a_\la^{\la'}-\Delta_1\ro,
x+a_\la^{\la'}+\Delta_1\ro] \subset [-r,r] - E_n(\la')\} \ge c_3\ro^{-(d+d')}
$$
and therefore
$$
\#\{\la', \om(\la)=\al(\la'), [x+a_\la^{\la'}-\Delta_1\ro,
x+a_\la^{\la'}+\Delta_1\ro] \subset F_\infty(\la')\} \ge c_3\ro^{-(d+d')}.
$$
Now we come back to $J_R$\,, setting, for $\und{a} \in \Sigma(\ro)$, $\und{a}' \in \Sigma'(\ro)$
$$
E^*(\und{a}, \und{a}') = \{e^x, x \in F_\infty(\und{a}, \und{a}',+1)\} \cup \{-e^x, x \in F_\infty(\und{a}, \und{a}', -1)\}.
$$
Observe that $F_\infty(\la)$ is contained in $V_{\Delta\ro}(F(\la))$; this implies conclusion (i) of the Scale recurrence Lemma. 

For conclusion (ii), as $\und{\te} \to k^{\und{\te}}$ is Lipschitz, there exists $C > 0$ such that if $\la = (\und{a}, \und{a}',u)$, $\la' = (\und{b}, \und{b}',v)$ satisfy $\om(\la) = \al(\la')$ and $\und{\te}$, $\und{\te'}$ are {\it any\/} elements in $\Sigma^-$, $\Sigma^{\prime -}$ ending with $\und{a}$, $\und{a}'$ respectively, then
$$
\lv\text{Log }|I^{\und{\te}'}(\und{b}')| - \text{ Log }|I^{\und{\te}}(\und{b})|
- a_\la^{\la'}\rv \le C\ro.
$$
Taking $\Delta_1 > 2C$, for $x \in F_\infty(\la)$, we will have
$$
x + \text{ Log }|I^{\und{\te}'}(\und{b}')| - \text{ Log
}|I^{\und{\te}}(\und{b})| + z \in F_\infty(\la')
$$
for $|z| \le \dfrac{\Delta_1}{2} \ro$ and at least $c_3\ro^{-(d+d')}$ elements $\la'$ (with $\al(\la') = \om(\la))$, for {\it all\/} $\und{\te}$, $\und{\te}'$ ending with $\und{a}$, $\und{a}'$ respectively. This immediately implies conclusion (ii) of the Scale recurrence Lemma.

\vglue .5in

\noindent{\bf 6.7}\quad We now prove the lemma at the end of \S6.5.

Let $\eta_0 > 0$ be extremely small, and $\xi \in [1, \Delta_0\,\ro^{-1}]$. If the norm of the operator in the lemma is $\ge (1-\eta_0)^{1/2}$, we can find $z \in \co^{\wedge}$ with
$$
\align
||z||^2 &= \Sigma\,p^\la|z_\la|^2 = 1,\\
||w||^2 &\ge 1-\eta_0\,,
\endalign
$$
where
$$
w_\la = \sum_{\la'} p_\la^{\la'}\,\exp(i\xi\,a_\la^{\la'})z_{\la'}\,.
$$
Now, we have
$$
|w_\la|^2 \le \sum_{\la'} p_\la^{\la'} |z_{\la'}|^2,
$$
and all $p^\la$ have order $\ro^{d+d'}$. Therefore there exists $\eta_1 > 0$ very small, and $\widetilde \La \subset \La$, with
$$
\align
&\qquad\#\, (\La-\widetilde\La) \le \eta_1\,\ro^{-(d+d')},\\
&|w_\la|^2 \ge (1-\eta_1) \sum_{\la'} p_\la^{\la'} |z_{\la'}|^2, \text {for all } \la \in \widetilde\La.
\endalign
$$
Let $\la \in \widetilde\La$,\,\,\, $\om(\la)=i$; for $\la'\in \La_i$\,, set
$$
Z_\la^{\la'} = \exp(i\,a_\la^{\la'}\,\xi)z_{\la'}\,.
$$
We have
$$
-|w_\la|^2 + \sum_{\La_i} p_\la^{\la'} |z_{\la'}|^2 = \frac 12 \sum_{\La_i} \sum_{\La_i} p_\la^{\la_0'}\,p_\la^{\la_1'} |Z_\la^{\la_0'} - Z_\la^{\la_1'}|^2,
$$
hence
$$
\sum_{\La_i} \sum_{\La_i} |Z_\la^{\la_0'} - Z_\la^{\la_1'}|^2 \le \eta_2\,\ro^{-2(d+d')}
$$
with very small $\eta_2 > 0$, as $p_\la^{\la'}$ has order $\ro^{d+d'}$. On the other hand $\sum\limits_{\La_i} |Z_\la^{\la'}|^2$ has order $\ro^{-(d+d')}$. Thus, if we write
$$
\align
&\qquad Z_\la = N_i^{-1}\,\sum_{\La_i} Z_\la^{\la'}\,,\\
&z_{\la'} = \exp(-i\,a_\la^{\la'}\,\xi)Z_\la + \widetilde Z_\la^{\la'}\,,
\endalign
$$
we will have $|Z_\la|$ of order 1 and
$$
\sum_{\La_i} |\widetilde Z_\la^{\la'}|^2 \le \eta_3\,\ro^{-(d+d')},
$$
with very small $\eta_3 > 0$.

If we take now $\la_0,\la_1 \in \widetilde\La \cap \La^i$, and compare the two expressions for $z_{\la'}$ that we get, we obtain
$$
\sum_{\La_i} |\exp(i\,a_{\la_0}^{\la'}\,\xi) - \exp(i\,a_{\la_1}^{\la'}\,\xi)|^2 \le \eta_4\,\ro^{-(d+d')},
$$
meaning that
$$
\lv \sin\bigg[\frac 12 (a_{\la_0}^{\la'} - a_{\la_1}^{\la'})\xi\bigg]\rv < \eta_5
$$
holds for all $\la'\in \La_i$ but at most $\eta_5\,\ro^{-(d+d')}$ elements. The constants $\eta_4$\,, $\eta_5$ here are still small.

To derive a contradiction, we will choose $\la_0$\,, $\la_1$ of the form
$$
\la_0 = (\und{a}^0, \und{a}', u), \qquad \la_1 = (\und{a}^1, \und{a}',u)
$$
which gives, with $\la'= (\und{b}, \und{b}',v)$:
$$
\align
a_{\la_0}^{\la '} &= \Log|I^{\und{\widetilde\te}^1}(\und{b})| - \Log|I^{\und{\widetilde\te}^0}(\und{b})|\\
&= \Log D\big[k^{\und{\widetilde\te}^1} \circ (k^{\und{\widetilde\te}^0})^{-1}\big](y)
\endalign
$$
where $\und{\widetilde\te}î$ ends with $\und{a}^i$ and $y \in I^{\und{\widetilde\te}^0}(\und{b})$.

It is now time to remember the hypothesis in the Scale recurrence Lemma: there exists $\und{\te}^0, \und{\te}^1 \in \Sigma^-$ (with the same last letter) and $x \in K^{\und{\te}_0}(\te_0^0)$, such that
$$
D\,\Log\,D[k^{\und{\te}^1}\circ(k^{\und{\te}^0})^{-1}](x) = \ga \ne 0.
$$
In a neighbourhood of $\und{\te}^0$\,, $\und{\te}^1$\,, $x$, this will still belongs to $\big[\dfrac 12 \ga, \dfrac 32 \ga\big]$. As $\#(\La-\widetilde\La) \le \eta_1\,\ro^{-(d+d')}$, we can find $\und{a}^0$, $\und{a}^1$, $\und{a}'$, $u$ such that $\la_i = (\und{a}^i, \und{a}',u) \in \widetilde\La$ and $\und{\widetilde\te}^i$ is close to $\und{\te}^i$. The number of $\la'= (\und{b}, \und{b}',v)$ such that 
$I^{\und{\widetilde\te}^0}(\und{b})$ is close to $x$ is still of order $\ro^{-(d+d')}$, and for two such $\la_0'$\,, $\la_1'$ we will have
$$
|(a_{\la_0}^{\la_0'} - a_{\la_1}^{\la_0'}) - (a_{\la_0}^{\la_1'} - a_{\la_1}^{\la_1'}) - \ga(y_0-y_1)| \le \frac 12 |\ga|\,|y_0-y_1|,
$$
where $y_i \in I^{\und{\widetilde\te}^0}(\und{b}^i)$,\,\,\, $\und{b}^i$ being the first component of $\la_1'$\,.

On the other hand, for most $\la'$ we should have
$$
|a_{\la_0}^{\la'} - a_{\la_1}^{\la'} - 2k\pi\xi^{-1}| \le \eta_6\,\xi^{-1}
$$
for some integer $k$ and small $\eta_6 > 0$. Combining the two inequalities, we should have for most choices of $\und{b}^0$\,, $\und{b}^1$ (with $I^{\und{\widetilde\te}^0}(\und{b}î)$ close to $x$) either $|y_0-y_1| \ge \pi\xi^{-1}|\ga|^{-1}$ or $|y_0-y_1| \le 4\eta_6\,\xi^{-1}|\ga|^{-1}$. We have here $1 \le \xi \le \Delta_0\ro^{-1}$, hence $\pi\xi^{-1}|\ga|^{-1} \ge \pi|\ga|^{-1}\,\Delta_0^{-1}\ro$. Taking $\Delta_0 \ll |\ga|$ and $\eta_6$ small enough contradicts the bounded geometry of regular Cantor sets.

\vglue .5in

\noindent{\bf 6.8}\quad We finally will prove the estimate
$$
\sum\,p^\la|V_{\Delta\ro}(\Hat{\Hat E}_n(\la))| \le \kappa_5\,\sum\,p^\la|V_{\Delta\ro}(E_n(\la))|
$$
of 6.6.

Recall that $\Hat{\Hat E}_n(\la)$ is the set of $x \in [-r, +r]$ such that the number of $\la'$, with $\om(\la) = \al(\la')$, for which $[x + a_\la^{\la'} - \Delta_1\ro, x+a_\la^{\la'} + \Delta_1\ro] \subset [-r, +r] - E_n(\la)$ is less than $c_3\,\ro^{-(d+d')}$.

Consider first the set $\widehat E_n(\la)$ introduced in 6.4: $\widehat E_n(\la)$ is the set of $x \in \re$ such that the number of $\la'$, with $\om(\la) = \al(\la')$, for which $[x + a_\la^{\la'} - \Delta_1\ro, x+a_\la^{\la'} + \Delta_1\ro] \subset \re - E_n(\la')$ is less than $(1-\tau)N_{\om(\la)}$\,.

As $N_{\om(\la)}$ is of the order of $\ro^{-(d+d')}$, we can choose $c_3$ small enough to have
$$
c_3\,\ro^{-(d+d')} < (1-\tau)N_{\om(\la)}\quad (\forall\, \la \in \La).
$$
Now, if $c_0 > 1$ is such that
$$
c_0^{-1} \ro \le |I^{\und{\te}}(\und{a})| \le c_0\ro
$$
for all $\und{\te} \in \Sigma^-$,\,\, $\und{a} \in \Sigma(\ro)$ with $\te_0 = a_0$\,, we will have
$$
|a_\la^{\la'}| \le 2\Log c_0
$$
for all $\la,\la'\in \La$ with $\om(\la) = \al(\la')$.

Then we will have
$$
\Hat{\Hat E}_n(\la) \cap [-r+2\Log c_0 + \Delta_1\ro, r-2 \Log c_0 - \Delta_1\ro] \subset \widehat E_n(\la).
$$

Now the corollary in 6.4 gives us an estimate of the desired form for $\sum\limits_\la p^\la|V_{\Delta\ro}(\widehat E_n(\la))|$, provided
$$
\underset{\la}\to{\text{Max }} |V_{\Delta\ro}(E_n(\la))|
$$
is sufficiently small.

This follows (inductively) from the desired estimate for $m < n$, as we have seen in 6.6, provided $\bar c_1$ is small enough.

We have still to control the $V_{\Delta\ro}$ neighbourhood of the part of $\Hat{\Hat E}_n(\la)$ which lies in
$$
[-r, -r+2 \Log c_0 + \Delta_1\ro] \cup [r - 2\Log c_0 - \Delta_1\ro, r].
$$
Let us consider for instance the part in $[r-2\Log c_0 - \Delta_1\ro, r]$.

The idea here is to take $r$ large enough, to choose a subinterval $J \subset [-r, +r]$ of length $5 \Log c_0$ where the $E_n(\la)$ are relatively thin, and to restrict to the set of $\la'$ which will send $[-r-2 \Log c_0 - \Delta_1\ro,r]$ into $J$.

To be more precise, given $\delta > 0$, we can find, if $r$ is large enough, a subinterval $J \subset [-r, +r]$ of length $5\Log c_0$ such that 
$$
\sum_\la p^\la |V_{\Delta\ro}(E_n(\la) \cap J)| \le \delta \sum_\la p^\la |V_{\Delta\ro}(E_n(\la))|.
$$

Consider now only those $\la'$ for which $r - \Log c_0 + a_\la^{\la'}$ belongs to the interval $J_1$ of same center as $J$ and length $2 \Log c_0$\,; the number of such $\la$'s is still of order $\ro^{-(d+d')}$, in particular much larger than $c_3\,\ro^{-(d+d')}$ if $c_3$ is small enough.

As all $p^\la$ have order $\ro^{(d+d')}$, we will obtain an estimate
$$
\sum_\la p^\la|V_{\Delta\ro}(\Hat{\Hat E}_n(\la) \cap [r-2\Log c_0 - \Delta\ro, r])| \le C\delta \sum_\la p^\la|V_{\Delta\ro}(E_n(\la))|.
$$
Taking $\delta > 0$ small enough, we will have $1 > \kappa_5 = 2C\delta + \kappa_4$ (where $\kappa_4$ is from 6.4), which concludes the proof.

\vglue .2in

\noindent{\bf 6.9}\quad In case A) of 6.5, we just set
$$
\align
&A = \AA \times \AA'\\
&\La = \Sigma(\ro) \times \Sigma'(\ro)\\
&F^{\pm}(\und{a}, \und{a}') = \{x, \pm e^x \in E(\und{a}, \und{a}')\}
\endalign
$$
and deal separately with the two families $(F^+(\und{a}, \und{a}'))$ and $(F^-(\und{a}, \und{a}'))$ in the same manner than in case B).


\newpage

\centerline{\bf REFERENCES}

\vglue .1in

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\vglue .3in

\noindent
 Carlos Gustavo T. de A. Moreira, Instituto de Matem\'atica Pura e Aplicada -- IMPA, Est. D. Castorina 110, 22460-320 Rio de Janeiro, Brasil 

\noindent {\it e-mail}: gugu\@impa.br

\vglue .1in

\noindent
J.-C. Yoccoz, Coll\`ege de France - 3, Rue d'Ulm F75005 - Paris - France


\end%