STABLE INTERSECTIONS OF REGULAR CANTOR SETS WITH LARGE HAUSDORFF DIMENSIONS

Carlos Gustavo T. de A. Moreira & Jean-Christophe Yoccoz

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.

Introduction

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 | x ∈ K_{1}, y ∈ 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.

This work has been partially supported by Pronex-Dynamical Systems, Brasil.

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) = [\sqrt{2}−1, 4(\sqrt{2}−1)]. It follows that the Lagrange spectrum L = \{lim sup_{\begin{matrix}p,q→∞ \\ p,q∈ℕ \end{matrix}} (q|qα−p|)^{−1}, α ∈ ℝ\ℚ\} contains a whole

half-line (for instance [6, +∞)). 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} ≅ 4, 52782956616 ….

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 (K̃_{1}, K̃_{2}) ∈ V ⇒ K̃_{1} ∩ K̃_{2} ≠ ∅. 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 ⊂ Ω^{∞} = {(K_{1}, K_{2}), K_{1} and K_{2} C^{k}-regular Cantor sets | HD(K_{1}) + HD(K_{2}) > 1} such that (K_{1}, K_{2}) ∈ U ⇒ ∃ t ∈ ℝ 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 §2.4) the following

Theorem: There is an open and dense set U ⊂ Ω^{∞} such that if (K_{1}, K_{2}) ∈ U, then I_{s}(K_{1}, K_{2}) is dense in K_{1} − K_{2} and HD((K_{1} − K_{2})\I_{s}(K_{1}, K_{2})) < 1, where I_{s}(K_{1}, K_{2}) := {t ∈ ℝ | K_{1} and (K_{2} + t) have stable intersection}.

We will also prove that the same result works if we replace stable intersection by dstable intersection, which is defined by asking that any pair (K̃_{1}, K̃_{2}) in some neighbourhood of (K_{1}, K_{2}) satisfies HD(K̃_{1} ∩ K̃_{2}) ≥ d: most pairs of Cantor sets (K_{1}, K_{2}) ∈ Ω^{∞} have d-stable intersection for any d < HD(K_{1}) + HD(K_{1}) − 1 (see section §2.6).

Moreover, we will show that the open set U mentioned in the theorem above is very large in Ω^{∞} in the sense that generic n-parameter families in Ω^{∞} are actually contained in U (see section §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]).

§1. Regular Cantor sets: the setting.

1.1 Let 𝔸 be a finite alphabet, B a subset of 𝔸^{2}, and Σ the subshift of finite type of 𝔸^{ℤ} with allowed transitions B.

We will always assume that Σ is topologically mixing, and that every letter in 𝔸 occurs in Σ.

An expansive map of type Σ is a map g with the following properties:

(i) the domain of g is a disjoint union ⋃_{B} I(a, b), where, for each (a, b), I(a, b) is a compact

subinterval of I(a) := [0, 1] × {a}; (ii) for each (a, b) ∈ B, the restriction of g to I(a, b) is a smooth diffeomorphism onto I(b)

satisfying |Dg(t)| > 1 for all t.

The regular Cantor set associated to g is the maximal invariant set

K = ⋂_{n≥0} g^{−n}( ⋃_{B} I(a, b)).

Let Σ^{+} be the unilateral subshift associated to Σ. There exists a unique homeomorphism h: Σ^{+} → K such that

h(\underline{a}) ∈ I(a_{0}), for \underline{a} = (a_{0}, a_{1}, …) ∈ Σ^{+}
h ∘ σ = g ∘ h.

Let r be a real number > 1, or r = +∞. The space of C^{r} expansive maps of type Σ, endowed with the C^{r} topology, will be denoted by Ω_{Σ}^{r}. The union Ω_{Σ} = ⋃_{r>1} Ω_{Σ}^{r} is endowed with the inductive limit topology. 1.2 For (a, b) ∈ B, let

f_{a,b} = [g|_{I(a,b)}]^{−1} ;

this is a contracting diffeomorphism from I(b) onto I(a, b). If \underline{a} = (a_{0}, …, a_{n}) is a word of Σ, we put

f\underline{a} = f_{a_{0},a_{1}} ∘ ⋯ ∘ 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(\underline{a}). 1.3 Let Σ^{−} = {(θ_{n})_{n≤0} , (θ_{i}, θ_{i+1}) ∈ B for i < 0}. We equip Σ^{−} with the following ultrametric distance: for \underline{θ} ≠ \underline{θ̃} ∈ Σ^{−}, set

d(\underline{θ}, \underline{θ̃}) = \{ \begin{matrix}1 if θ_{0} ≠ θ̃_{0} \\ |I(\underline{θ} ∧ \underline{θ̃})| otherwise \end{matrix}

where \underline{θ} ∧ \underline{θ̃} = (θ_{−n}, …, θ_{0}) if θ̃_{−j} = θ_{−j} for 0 ≤ j ≤ n and θ̃_{−n−1} ≠ θ_{−n−1} .

Now, let \underline{θ} ∈ Σ^{−}; for n > 0, let \underline{θ}^{n} = (θ_{−n}, …, θ_{0}), and let B(\underline{θ}^{n}) be the affine map from I(\underline{θ}^{n}) onto I(θ_{0}) such that the diffeomorphism k\underline{θ} ; n = B(\underline{θ}^{n})∘f\underline{θ}^{n} is orientation preserving.

We have the following well-known result (see [Su]): Proposition. Let r ∈ (1, +∞), g ∈ Ω_{Σ}^{r}.

1. For any \underline{θ} ∈ Σ^{−}, there is a diffeomorphism k\underline{θ} ∈ Diff_{+}^{r}(I(θ_{0})) such that k\underline{θ} ; n converge to k\underline{θ} in Diff_{+}^{r′}(I(θ_{0})), for any r′ < r, uniformly in \underline{θ}. The convergence is also uniform in a neighbourhood of g in Ω_{Σ}^{r}.

2. If r is an integer, or r = +∞, k\underline{θ} ; n converge to k\underline{θ} in Diff_{+}^{r}(I(θ_{0})). More precisely, for every 0 ≤ j ≤ r − 1, there is a constant C_{j} (independent on \underline{θ} such that

|D^{j} Log D[k\underline{θ} ; n ∘ (k\underline{θ})^{−1}](x)| ≤ C_{j}|I(\underline{θ}^{n})|.

It follows that \underline{θ} → k\underline{θ} is Lipschitz in the following sense: for θ_{0} = θ̃_{0} , we have

|D^{j} Log D[k\underline{θ̃} ∘ (k\underline{θ})^{−1}](x)| ≤ C_{j} d(\underline{θ}, \underline{θ̃}).

1.4 For \underline{θ} ∈ Σ^{−} and \underline{a} a word in Σ starting with a_{0} = θ_{0} , we denote:

K(\underline{a}) = K ∩ I(\underline{a}),

I\underline{θ}(\underline{a}) = k\underline{θ}(I(\underline{a})),
K\underline{θ}(\underline{a}) = k\underline{θ}(K(\underline{a})).

The different K\underline{θ} are related in the following way: let F \underline{θ} be the affine map from I(θ_{0}) onto I^{σ^{−1}(\underline{θ})}(θ_{−1}, θ_{0}) with the same orientation than f_{θ_{−1}θ_{0}} . Then, on I(θ_{0}):

F \underline{θ} ∘ k\underline{θ} = k^{σ^{−1}(\underline{θ})} ∘ f_{θ_{−1}θ_{0}} .

If we define

I = \{(\underline{θ}, x), \underline{θ} ∈ Σ^{−}, x ∈ I(θ_{0})\}
K = \{(\underline{θ}, x), \underline{θ} ∈ Σ^{−}, x ∈ K\underline{θ}(θ_{0})\}
F (\underline{θ}, x) = (σ^{−1}(\underline{θ}), F \underline{θ}(x)),

we have a fiber-contracting map F such that

K = ⋂_{n≥0} F^{n}(I).

Moreover, if we define H: Σ → K by

H(\underline{θ}) = (π^{−}(\underline{θ}), k^{π^{−}(\underline{θ})} ∘ h(π^{+}(\underline{θ})))

where π^{−}, π^{+} are the natural projections of Σ onto Σ^{−}, Σ^{+} respectively, the map H is an homeomorphism which satisfies

H ∘ σ^{−1} = F ∘ H.

1.5 Let r ∈ (1, +∞]. For a ∈ 𝔸, denote by P^{r}(a) the space of C^{r}-embeddings of I(a) into ℝ, endowed with the C^{r} topology. The affine group Aff(ℝ) acts by composition on the left on P^{r}(a), the quotient space being denoted by \overline{P}^{r}(a). We also consider P(a) = ⋃_{r>1} P^{r}(a)

and \overline{P}(a) = ⋃_{r>1} \overline{P}^{r}(a), endowed with the inductive limit topologies.

Let (a_{0}, a_{1}) ∈ B, and g ∈ Ω_{Σ}^{r}. We define the renormalization operator
T_{a_{1}a_{0}}^{g}: P^{r}(a_{0}) → P^{r}(a_{1})
h ↦ h ∘ f_{a_{0},a_{1}} .

If \underline{a} = (a_{0}, …, a_{n}) is a word in Σ, we write T g ; \underline{a}: P^{r}(a_{0}) → P^{r}(a_{n}) for T_{a_{n}a_{n−1}}^{g} ∘ ⋯ ∘ T_{a_{1},a_{0}}^{g} .

We have

T g ; \underline{a}(h) = h ∘ f\underline{a} .

The renormalization operators commute to the action of the affine group, hence they are defined on the quotient spaces \overline{P}^{r}(a) as well.

We can reinterpret the proposition above as follows. For \underline{θ} ∈ Σ^{−}, the sequence T_{\underline{θ}^{n}}^{g}(h_{n}) (h_{n} ∈ P^{r}(θ_{−n})) converge (in the C^{r′} topology, r′ < r) to a limit in \overline{P}^{r}(θ_{0}), provided the h_{n} stay bounded. The limit is independent of the h_{n} and the convergence is uniform in \underline{θ} 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 = \{(\underline{θ}, A)\}, where \underline{θ} ∈ Σ^{−} and A is now an affine embedding of I(θ_{0}) into ℝ. We have a canonical map

A → P^{r} = ⋃_{𝔸} P^{r}(a)
(\underline{θ}, A) ↦ A ∘ k\underline{θ} (∈ P^{r}(θ_{0})).

We can lift the action of the renormalization operators to A: if (θ_{0}, θ_{1}) ∈ B, we set

T_{θ_{1},θ_{0}}(\underline{θ}, A) = (\underline{θ}θ_{1}, A ∘ F^{\underline{θ}θ_{1}}).

At this level, the various T_{θ_{1},θ_{0}} are just the inverse branches of the map S:

S(\underline{θ}, A) = (σ^{−1}(\underline{θ}), A ∘ (F \underline{θ})^{−1}).

The affine group Aff(ℝ) still acts by composition on the left on A (fiberwise onto Σ^{−}).

§2. Intersections of regular Cantor sets: setting and statement of the main result.

2.1 In this section, we assume we are given two sets of data (𝔸, B, Σ, g), (𝔸′, B′, Σ′, g′) defining regular Cantor sets K, K′.

We define as in §1.5 the spaces P = ⋃_{𝔸} P(a) and P′ = ⋃_{𝔸′} P(a′).

A pair (h, h′), (h ∈ P(a), h′ ∈ P′(a′)) is called a smooth configuration for K(a), K′(a′). Actually, rather than working in the product P × P′, it is better to go to the quotient Q by the diagonal action of the affine group Aff(ℝ). Elements of Q are called smooth relative configurations for K(a), K′(a′).

We say that a smooth configuration (h, h′) ∈ P(a) × P(a′) is

– linked if h(I(a)) ∩ h′(I(a′)) ≠ ∅;

– intersecting if h(K(\underline{a})) ∩ h′(K(\underline{a}′)) ≠ ∅; – stably intersecting if it is still intersecting when we perturb it in P ×P′, and we perturb

(g, g′) in Ω_{Σ} × Ω_{Σ′} . 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}), 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.

2.2 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 × A′ by the diagonal action on the left of the affine group. An element of C, represented by (\underline{θ}, A) ∈ A, (\underline{θ}′, A′) ∈ A′, is called a relative configuration of the limit geometries determined by \underline{θ}, \underline{θ}′. We have canonical maps

A × A′ → P × P′
C → Q

which allow to define linked, intersecting, and stably intersecting configurations at the level of A × A′ or C.

Remark: For a configuration ((\underline{θ}, A), (\underline{θ}′, A′)) of limit geometries, one could also consider the weaker notion of stable intersection, obtained by considering perturbations of g, g′ in Ω_{Σ} × Ω_{Σ′} and perturbations of (\underline{θ}, A), (\underline{θ}′, A′) in A × A′. We do not know of any example of expansive maps g, g′, and configurations (\underline{θ}, A), (\underline{θ}′, A′) which are stably intersecting in the weaker sense but not in the stronger sense.

2.3 When looking for stably intersecting configurations, the following notion is crucial.

A compact set L in C is recurrent if for every u ∈ L we can find v ∈ int L and ℓ, ℓ′ ≥ 0 with ℓ + ℓ′ > 0 and u = S^{ℓ}S^{′ℓ′}(v). In other terms, if (\underline{θ}, A), (\underline{θ}′, A′) represents u, we can find words \underline{a} = (a_{0}, …, a_{ℓ}), \underline{a}′ = (a′_{0}, …, a′_{ℓ′}) in Σ, Σ′ respectively, with a_{0} = θ_{0} , a′_{0} = θ′_{0} , such that (T\underline{a}(\underline{θ}, A), T′_{\underline{a}′}(\underline{θ}′, A′)) = v belongs to int L.

We say that L is immediately recurrent if moreover we can take ℓ + ℓ′ = 1 (for any u ∈ L).

Let L be a recurrent compact set. Then one can find finitely many compact sets L_{1}, …, L_{N} , and words \underline{a}^{i}, \underline{a}^{′i}, 1 ≤ i ≤ N (with, for every i, at least one having more than one letter) such that

(i) ⋃_{1≤i≤N} L_{i} is a neighbourhood of L;

(ii) T\underline{a}^{i} T′_{\underline{a}^{′i}} is defined on L_{i} and sends L_{i} into int L. From this, we deduce immediately the following properties of recurrent compact sets.

1. Any recurrent compact set is contained in an immediately recurrent compact set.

2. Any recurrent (resp. immediately recurrent) compact set for g, g′ is still recurrent (resp. immediately recurrent) for g̃, g̃′ in a neighbourhood of (g, g′) in Ω_{Σ} × Ω_{Σ′} .

3. Any relative configuration contained in a recurrent compact set is intersecting.

The last property can be strengthened:

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

Proof: By property 1 above, we may assume that the given configuration u = [(\underline{θ}, A), (\underline{θ}′, A′)] is contained in an immediately recurrent compact set L. We first show that any small perturbation of (A∘k\underline{θ}, A′ ∘k^{′\underline{θ}′}) in P(θ_{0})×P′(θ′_{0}) is still an intersecting smooth configuration for (g, g′). Such a perturbation can be written (h ∘ A ∘ k\underline{θ}, h′ ∘ A′ ∘ k^{′\underline{θ}′}), 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 in Q (h̃ ∘ Ã∘ k\underline{θ̃}, h̃′ ∘ Ã′ ∘ k^{′\underline{θ̃}′}), with h̃, h̃′ closer to the identity than were h, h’, and [(Ã, θ̃), (Ã′, θ̃′)] 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 Ω_{Σ} × Ω_{Σ′}

(see property 2 above); this concludes the proof. □

2.4 Our main result is now the following.

THEOREM: Let g, g′ be two C^{∞} expansive maps as above. Assume that the associated Cantor sets satisfy

HD(K) + HD(K′) ≥ 1.

Then, we can find, arbitrarily close to g, g′ (in the C^{∞}-topology), perturbations g̃, g̃′ having a non empty recurrent compact set of relative configurations.

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 Ω_{Σ} × Ω_{Σ′} . A pair (g, g′) belongs to V if for any [(\underline{θ}, A), (\underline{θ}′, A′)] ∈ A × A′ there is a translation R_{t} (in ℝ) such that (R_{t} ∘ A ∘ k\underline{θ}, A′ ∘ k^{′\underline{θ}′}) is a stably intersecting configuration.

THEOREM:

1. V is open in Ω_{Σ} × Ω_{Σ′} , and V ∩ (Ω_{Σ}^{∞} × Ω_{Σ′}^{∞}) is dense (for the C^{∞}-topology) in the set {(g, g′), HD(K) + HD(K′) > 1}.

2. Let (g, g′) ∈ V. There exists d^{∗} < 1 such that for any (h, h′) ∈ P × P′, the set

I_{s} = {t ∈ ℝ, (R_{t} ∘ h, h′) is a stably intersecting smooth configuration for (g, g′)}

is (open and) dense in

I = {t ∈ ℝ, (R_{t} ∘ h, h′) is an intersecting smooth configuration for (g, g′)}

and moreover HD(I − I_{s}) ≤ d^{∗}. The same d^{∗} is also valid for (g̃, g̃′) in a neigbourhood of (g, g′) in Ω_{Σ} × Ω_{Σ′} .

Proof: Let us first prove that V is open. For λ ∈ ℝ^{∗}, denote by H_{λ} the homothety x ↦ λx. Let (g, g′) ∈ Ω_{Σ} × Ω_{Σ′} , 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 \underline{θ} ∈ Σ^{−}, \underline{θ}′ ∈ Σ^{′−}, λ ∈ J_{R} = [−R, −R^{−1}] ∪ [R^{−1}, R], there exists t ∈ ℝ such that (R_{t} ∘ k\underline{θ}, H_{λ} ∘ k^{\underline{θ}′}) is a stably intersecting configuration. Using the compactness of the set of (\underline{θ}, \underline{θ}′, λ) under consideration, we see not only that V is open but for (g, g′) ∈ V the following is true: there exist δ > 0, a neighbourhood W of (g, g′) in Ω_{Σ} × Ω_{Σ′} and a neighbourhood Z of the identity in ⋃_{r>1} Diff^{r}(ℝ) such that for any (\underline{θ}, \underline{θ}′, λ) ∈ Σ^{−} × Σ^{′−} × J_{R} there is an interval J of length

δ with (R_{t} ∘ h̃ ∘ k\underline{θ}, h̃′ ∘ k^{\underline{θ}′}) being a stably intersecting configuration for all t ∈ J, h̃, h̃′ ∈ Z and (g̃, g̃′) ∈ W.

Assertion 2 in the theorem is now an easy consequence of the last statement. Let (g, g′) ∈ V, (h, h′) ∈ P ×P′, t ∈ I. Let ε > 0 be small; let \underline{a}, \underline{a}′ be words in Σ, Σ′ respectively such that I(\underline{a}), I(\underline{a}′) have size of order ε, and such that h(K(\underline{a})) ∩ h′(K(\underline{a}′)) ≠ ∅. We apply (in the quotient space Q) T\underline{a}T′_{\underline{a}′} to (R_{t} ∘ h, h′) and we get an intersecting configuration of the form (h̃ ∘ k\underline{θ}, h̃′ ∘ H_{λ} ∘ k^{\underline{θ}′}), where \underline{θ}, \underline{θ}′ end with \underline{a}, \underline{a}′ respectively and h̃, h̃′ ∈ Z if ε is small enough. The statement above now shows that there is an interval of length cδε (for some fixed c > 0) contained in I_{s} ∩ [t − ε, t + ε]. 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 cδ (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^{′n′}(x′)|}

is irrational, then (g, g′) ∈ V. Indeed, let \underline{a} (resp. \underline{a}′) the word in Σ (resp. Σ′) of length n + 1 (resp. n′ + 1) such that f\underline{a}(x) = x (resp. f′_{\underline{a}′}(x′) = x′). Let N be the set of (\underline{θ}, \underline{θ}′, λ) ∈ Σ^{−}×Σ^{′−}×ℝ^{∗} such that there exists t ∈ ℝ for which (R_{t}∘k\underline{θ}, H_{λ}∘k^{\underline{θ}′}) 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 (\underline{θ}, \underline{θ′}) ∈ Σ^{−} × Σ^{′−} there are λ_{1} < 0 < λ_{2} such that (\underline{θ}, \underline{θ′}, λ_{i}) ∈ N for i = 1, 2. But then, setting \underline{ā} = (… \underline{a}, \underline{a}, \underline{a}) and \underline{ā}′ = (… \underline{a}′, \underline{a}′, \underline{a}′), the hypothesis on the eigenvalues imply that (\underline{ā}, \underline{ā}′, λ) ∈ N for all λ ∈ ℝ^{∗}. Using again the renormalization operators, we finally get that N = Σ^{−} × Σ^{′−} × ℝ^{∗}. The proof is complete. □

2.5 Let d̃> 0.

A smooth configuration (h, h′) ∈ P(a)×P(a′) has d̃-stable intersection if the Hausdorff dimension of h(K(a)) ∩ h′(K(a′)) is > d̃, and this property still holds when we perturb (h, h′) ∈ P(a) × P(a′) and (g, g′) in Ω_{Σ} × Ω_{Σ′} .

In relation to this property, we say that a compact subset L ⊂ C is d̃-recurrent if the following holds: for any ((\underline{θ}, A), (\underline{θ}′, A′)) representing a point u ∈ L, we can find pairs (\underline{a}^{1}, \underline{a}^{′1}), …, (\underline{a}^{k}, \underline{a}^{′k}) for words such that

i) the first letter of \underline{a} (resp. \underline{a}^{′i}) is θ_{0} ; ii) the intervals I(\underline{a}^{i}) are disjoint; iii) (T\underline{a}^{i}(\underline{θ}, A), T′_{\underline{a}^{′i}}(\underline{θ}′, A′)) represents a point in int L for 1 ≤ i ≤ k;

iv) ∑_{1}^{k} |I\underline{θ}(\underline{a}^{i})|d̃ > 1.

By the compactness of L, we can find the \underline{a}^{i}, \underline{a}^{′i} with uniformly bounded length,

iv’) ∑_{1}^{k} |I\underline{θ}(\underline{a}^{i})|d̃′ ≥ c > 1,

with d̃′ slightly larger than d̃.

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

Proposition: If a relative affine configuration belongs to a d̃-recurrent non empty compact subset of C, it is a d̃-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.

THEOREM: Let 0 < d̃ < HD(K) + HD(K′) − 1. Then we can find arbitrarily small C^{∞}-perturbations of (g, g′) which exhibit a non empty d̃-recurrent compact subset.

§3. Outline of the proof of the main theorem.

3.1 Let g, g′ be expansive maps as above. If ((\underline{θ}, A), (\underline{θ}′, A′)) is a relative configuration, the affine map A^{−1} ∘ A′: ℝ × {θ′_{0}} → ℝ × {θ_{0}} is well defined; the coefficient of x is a number in ℝ^{∗} which is called the relative scale of the given relative configuration. We obtain thus the space S = Σ^{−} × Σ^{′−} × ℝ^{∗} 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 S.

Actually, let \underline{θ} ∈ Σ^{−}, \underline{θ}′ ∈ Σ^{′−}, and \underline{a} be a word in Σ starting with θ_{0} . Let
T\underline{a} = T_{a_{n},a_{n−1}} ∘ ⋯ ∘ T_{a_{1},a_{0}} .

Then we have, for s ∈ ℝ^{∗}

T\underline{a}(\underline{θ}, \underline{θ}′, s) = (\underline{θ}a_{1} … a_{n}, \underline{θ}′, εs|I\underline{θ}(\underline{a})|^{−1})

where ε = +1 (resp. −1) if f\underline{a} is orientation preserving (resp. reversing).

Similarly, if \underline{a}′ is a word in Σ starting with θ′_{0} , we have
T′_{\underline{a}′}(\underline{θ}, \underline{θ}′, s) = (\underline{θ}, \underline{θ}′a′_{1} … a′_{n}, ε′s|I^{\underline{θ}′}(\underline{a}′)|).

3.2 The first main ingredient in the proof of the main theorem is a recurrence result for the action of renormalization on the space S 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

S_{R} = Σ^{−} × Σ^{′−} × ([−R, −R^{−1}] ∪ [R^{−1}, R])

of S, with R conveniently large.

Fix once and for all a conveniently large constant c_{0} . The size of a word \underline{a} in Σ is the length |I(\underline{a})|; we say that \underline{a} has approximate size ρ if

c_{0}^{−1} ρ ≤ |I(\underline{a})| ≤ c_{0}ρ

and denote by Σ(ρ) the set of those words. Its cardinality is of order ρ^{−d}, where we set

d = HD(K), d′ = HD(K′).

We define similarly Σ′(ρ), with cardinality of order ρ^{−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 \underline{θ}^{0}, \underline{θ}^{1} ∈ Σ^{−}, with θ_{0}^{0} = θ_{0}^{1} , and x_{0} ∈ K^{\underline{θ}^{0}}(θ_{0}^{0}) such that

|D Log D(k^{\underline{θ}^{1}} ∘ (k^{\underline{θ}^{0}})^{−1})(x_{0})| ≠ 0.

Then, denoting J_{R} = [−R, −R^{−1}] ∪ [R^{−1}, R], we have:

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 < ρ < 1, and a family E(\underline{a}, \underline{a}′) of subsets of J_{R}, (\underline{a}, \underline{a}′) ∈ Σ(ρ) × Σ′(ρ), such that

m(J_{R} − E(\underline{a}, \underline{a}′)) ≤ c_{1} , ∀ (\underline{a}, \underline{a}′)

we can find another family E^{∗}(\underline{a}, \underline{a}′) of subsets of J_{R} satisfying:

(i) for any (\underline{a}, \underline{a}′), E^{∗}(\underline{a}, \underline{a}′) is contained in a c_{2}ρ-neighbourhood of E(\underline{a}, \underline{a}′); (ii) let (\underline{a}, \underline{a}′) ∈ Σ(ρ) × Σ′(ρ), s ∈ E^{∗}(\underline{a}, \underline{a}′); we can find at least c_{3} ρ^{−(d+d′)} pairs (\underline{b}, \underline{b}′) ∈

Σ(ρ) × Σ′(ρ) (with \underline{b}, \underline{b}′ starting respectively with the last letter of \underline{a}, \underline{a}′) such that, if \underline{θ} ∈ Σ^{−}, \underline{θ}′ ∈ Σ^{′−} end respectively with \underline{a}, \underline{a}′ and

T\underline{b}T′_{\underline{b}′}(\underline{θ}, \underline{θ}′, s) = (\underline{θ̃}, \underline{θ̃}′, s̃)

the ρ-neighbourhood of s̃ is contained in E^{∗}(\underline{b}, \underline{b}′).

Observe that \underline{θ̃}, \underline{θ̃}′ above end respectively with \underline{b}, \underline{b}′.

Let S^{∗}(\underline{a}, \underline{a}′) be the set of (\underline{θ}, \underline{θ}′, s) ∈ S_{R} such that \underline{θ} ends with \underline{a}, \underline{θ}′ with \underline{a}′ and s ∈ E^{∗}(\underline{a}, \underline{a}′); set S^{∗} = ⋃_{\underline{a},\underline{a}′} S^{∗}(\underline{a}, \underline{a}′). What property (ii) says is that there is a positive

proportion of compatible (\underline{b}, \underline{b}′) ∈ Σ(ρ) × Σ′(ρ) such that the image by the inverse branch T\underline{b}T\underline{b}′ is “well inside” S^{∗}.

3.3 The fibers of the quotient map: C → S are one-dimensional and have a canonical affine structure; moreover this bundle map is trivializable; we choose now an explicit trivialization C ≅ S × ℝ in order to have a coordinate in each fiber.

For each a ∈ 𝔸 (resp. a′ ∈ 𝔸′), we choose ω(a) ∈ Σ^{+} starting with a (resp. ω(a′) ∈ Σ^{′+} starting with a′). Given a configuration represented by (\underline{θ}, A), (\underline{θ}′, A′), we normalize (by the action of the affine group) in order to obtain that A′: ℝ × {θ_{0}} → ℝ becomes a translation sending k^{′\underline{θ}′}(h′(ω(θ′_{0}))) to 0.

Then our coordinate t((\underline{θ}, A), (\underline{θ}′, A′)) on the fiber will be by definition

t = A(k\underline{θ}(h(ω(θ_{0})))).

With the chosen normalization, A′(I(θ′_{0})) has size 1 while A(I(θ_{0})) has size |s|.

3.4 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^{∞}-topology in order to create a non empty recurrent compact set of relative configurations.

But a neighbourhood in the C^{∞} topology is a neighbourhood in some C^{k} topology for finite k. We now fix such an integer k ≥ 2.

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

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

3.5 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 Σ_{0} of Σ(ρ^{1/k}) such that

K = ⋃_{\underline{a}∈Σ_{0}} K(\underline{a})

is a partition of K into disjoint cylinders.

We then define Σ_{1} as the subset of Σ_{0} formed of the words \underline{a} ∈ Σ_{0} such that no word in Σ(ρ^{1/3k}) appears twice in \underline{a}.

Let c_{4} > 1 be a constant sufficiently close to 1 to have the following: let Î(\underline{a}), for \underline{a} ∈ Σ_{0} , be the interval with same center as I(\underline{a}) and c_{4}^{2} times the size of I(a); then the Î(\underline{a}), \underline{a} ∈ Σ_{0} , are pairwise disjoint.

We then choose, once and for all, a smooth even function χ: ℝ → ℝ satisfying χ(x) = 1 for |x| ≤ c_{4} ,

χ(x) = 0 for |x| ≥ c_{4}^{2} .

For \underline{a} ∈ Σ_{1} , we define the vectorfield X\underline{a}, with support ⊂ Î(\underline{a}) by

X\underline{a}(x) = c_{5} ρ^{(1+1/2k)} χ(B(x))\frac{∂}{∂x}

where c_{5} > 0 is a conveniently large constant, to be chosen later, and B is an affine map sending I(\underline{a}) onto [−1, +1]; there are two such maps, but as χ is even, they give the same X\underline{a}.

The probability space underlying the famility of random perturbations is Ω = [−1, 1]^{Σ_{1}}, equipped with the normalized Lebesgue measure.

For \underline{ω} = (ω(\underline{a}))_{\underline{a}∈Σ_{1}} ∈ Ω, we define Φ\underline{ω} to be the time-one map of the vector field X\underline{ω} = − ∑_{\underline{a}} ω(\underline{a})X\underline{a}, and g\underline{ω} to be g ∘ Φ\underline{ω}. Our task will be to find \underline{ω} ∈ Ω such that (g\underline{ω}, g′)

have a non empty compact recurrent set of relative configurations.

3.6 We make now some easy remarks on the family g\underline{ω}; the finer properties will be studied later.

3.6.1 By construction, the vectorfields X\underline{a}, \underline{a} ∈ Σ_{1} , have disjoint supports.

The intervals I(\underline{a}), \underline{a} ∈ Σ_{1} , have approximate size ρ^{1/k}; therefore the affine map B entering the definition of X\underline{a} has derivative of the order of ρ^{−1/k}. It follows that, when ρ is small, X\underline{a} is small in the C^{k}-topology. Then, the Φ\underline{ω}, for any \underline{ω} ∈ Ω, are close to the identity in the C^{k}-topology.

In particular, the estimates in proposition 1.3 for the k\underline{θ} are valid for g\underline{ω} uniformly in \underline{ω}, if we consider no more than k derivatives.

3.6.2 For a ∈ 𝔸, let

x_{min}(a) = inf K(a),
x_{max}(a) = sup K(a).

The finite set ⋃_{a} \{x_{min}(a), x_{max}(a)\} is invariant under g. Let x_{0} be a point in this set; if \underline{a}_{0} is

the element of Σ_{0} such that x_{0} ∈ K(\underline{a}_{0}), \underline{a}_{0} is the initial part of an eventually periodic word in Σ^{+}. Therefore \underline{a}_{0} ∈ Σ_{0} − Σ_{1} if ρ is small enough. It follows that g\underline{ω} coincides with g in I(\underline{a}_{0}) (for any \underline{ω} ∈ Ω), and that the points x_{min}(a), x_{max}(a) are still the “endpoints” of the perturbed Cantor set K\underline{ω}(a).

3.6.3 For \underline{a} ∈ Σ_{0} , let Ĩ(\underline{a}) be the ρ-neighbourhood of I(\underline{a}).

Let \underline{a} ∈ Σ_{0} , a_{−1} ∈ 𝔸 such that (a_{−1}, a_{0}) ∈ B; let \underline{a}′ be the initial part of a_{−1}\underline{a} that belongs to Σ_{0} . We have f_{a_{−1}a_{0}}(I(\underline{a})) ⊂ I(\underline{a}′); because the f_{a,b} are contractions, there exists c_{6} > 0 such that the c_{6}ρ-neighbourhood of f_{a_{−1}a_{0}}(Ĩ(\underline{a})) is contained in Ĩ(\underline{a}′).

On the other hand, for the perturbed inverse branch f\underline{ω} ; a_{−1}a_{0} , \underline{ω} ∈ Ω, we will have, for any x ∈ Ĩ(\underline{a})

f_{a_{−1}a_{0}}(x) = \{ \begin{matrix}f_{a_{−1}a_{0}}(x) if \underline{a}′ ∈ Σ_{0} − Σ_{1} \\ f_{a_{−1}a_{0}}(x) + c_{5}ρ^{1+1/2k} ω(\underline{a}′) if \underline{a}′ ∈ Σ_{1} \end{matrix}

(because X\underline{a}′ is constant on Ĩ(\underline{a}′)). This allows to conclude that f\underline{ω} ; a_{−1}a_{0}(Ĩ(\underline{a})) ⊂ Ĩ(\underline{a}′) (if ρ is small enough), and therefore that the perturbed Cantor set K\underline{ω} will be contained in ⋃_{Σ_{0}} Ĩ(\underline{a}).

3.7 We are now ready to give some information on the set L = L\underline{ω} of relative configurations for which we want to prove recurrence for at least one \underline{ω} ∈ Ω.

As indicated, this set will depend on \underline{ω}, but only as far as the fiber coordinate t is concerned. The image of L\underline{ω} under the projection map: C → S will be a subset L̃ of S independent of \underline{ω}.

Let Σ_{nr}(ρ^{3}) be the subset of Σ(ρ^{3}) formed by words \underline{a} such that:

(i) no word \underline{b} ∈ Σ(ρ^{1/3k}) appears twice in \underline{a}; (ii) if \underline{c} ∈ Σ(ρ^{1/6k}) appears at the end of \underline{a}, then it does not appear elsewhere in \underline{a}.

We next define Σ_{nr}^{−} as the subset of Σ^{−} formed by \underline{θ} which end with a word in Σ_{nr}(ρ^{3}). This is an open and closed subset in Σ^{−}.

To define L̃, we will apply the Scale recurrence Lemma (with ρ^{1/2} instead of ρ). A family of subsets E(\underline{a}, \underline{a}′) of J_{R}, for (\underline{a}, \underline{a}′) ∈ Σ(ρ^{1/2}) × Σ′(ρ^{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(\underline{a}, \underline{a}′)) ≤ c_{1} , ∀(\underline{a}, \underline{a}′).

Then, the Scale recurrence Lemma gives us another family E^{∗}(\underline{a}, \underline{a}′) ∈ Σ(ρ^{1/2}) × Σ′(ρ^{1/2}), with the properties (i), (ii) indicated in the statement of the lemma.

The set L̃ is defined to be the subset of S_{R} formed by the (\underline{θ}, \underline{θ}′, s) such that \underline{θ} ∈ Σ_{nr}^{−}, and there exists \underline{a} ∈ Σ(ρ^{1/2}), \underline{a}′ ∈ Σ′(ρ^{1/2}) with s ∈ E^{∗}(\underline{a}, \underline{a}′) and \underline{θ}, \underline{θ}′ ending with \underline{a}, \underline{a}′ respectively.

3.8 For every (\underline{θ}, \underline{θ}′, s) in L̃, we will define in section 4, in relation to the conclusions of Marstrand’s theorem, a non empty subset L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s), depending on \underline{ω} ∈ Ω, of the fiber of C over (\underline{θ}, \underline{θ}′, s).

Let

L0 ; \underline{ω} = \{(\underline{θ}, \underline{θ}′, s, t), (\underline{θ}, \underline{θ}′, s) ∈ L̃, t ∈ L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s)\};

consider next the ρ-neighbourhood L1 ; \underline{ω} of L0 ; \underline{ω} in L̃ × ℝ:

L1 ; \underline{ω} = \{(\underline{θ}, \underline{θ}′, s, t), (\underline{θ}, \underline{θ}′, s) ∈ L̃ and ∃ (\underline{θ}_{0}, \underline{θ}′_{0}, s_{0}, t_{0}) ∈ L0 ; \underline{ω}

with d(\underline{θ}, \underline{θ}_{0}) < 2ρ^{5/2}, d(\underline{θ}′, \underline{θ}′_{0}) < 2ρ^{5/2}, |s − s_{0}| < ρ, |t − t_{0}| < ρ\}.

Fix u = (\underline{θ}, \underline{θ}′, s, t) ∈ L̃ × ℝ. We define two subsets Ω^{0}(u), Ω^{1}(u) of Ω. First,

Ω^{1}(u) = \{\underline{ω} ∈ Ω, (\underline{θ}, \underline{θ}′, s, t) ∈ L1 ; \underline{ω}\}.

Second, Ω^{0}(u) is the set of \underline{ω} ∈ Ω such that there exists \underline{b} ∈ Σ(ρ), \underline{b}′ ∈ Σ′(ρ), with b_{0} = θ_{0} , b′_{0} = θ′_{0} and the image T\underline{b}T′_{\underline{b}′}(u) = (\underline{θ̃}, \underline{θ̃}′, s̃, t̃) satisfies:

(i) for any s̃_{1} with |s̃ − s̃_{1}| < \frac{1}{2}ρ^{1/2}, we have (\underline{θ̃}, \underline{θ̃}′, s̃_{1}) ∈ L̃;

(ii) t̃∈ L0 ; \underline{ω}(\underline{θ̃}, \underline{θ̃}′, s̃).

[The renormalization operators T\underline{b}, T\underline{b}′ depend on \underline{ω}; the dependence is not indicated in the notations to keep the text readable].

The following crucial estimate will be proven in section 5.

Proposition: Assume that c_{5} is chosen conveniently large. Then there exists c_{7} > 0, such that, for any u ∈ L̃ × ℝ, one has

ℙ(Ω^{1}(u) − Ω^{0}(u)) ≤ exp ( − c_{7}ρ^{− \frac{1}{2k} (d+d′−1)}).

Basically, the reason for this inequality is the following: the sets L(\underline{θ}, \underline{θ}′, s) are formed of values of t such that T\underline{b}T′_{\underline{b}′}(\underline{θ}, \underline{θ}′, s, t) is bounded for many pairs (\underline{b}, \underline{b}′) ∈ Σ(ρ)×Σ′(ρ). It will follow from Marstrand’s theorem that the Lebesgue measure of L(\underline{θ}, \underline{θ}′, s) is uniformly bounded from below. So we can expect for fixed (\underline{b}, \underline{b}′) the probability (in \underline{ω}) that T\underline{b}T′_{\underline{b}′}(\underline{θ}, \underline{θ}′, s, t) is right to be bounded from below. When (\underline{b}, , \underline{b}′) vary, we will be able to find c ρ^{− \frac{1}{2k} (d+d′−1)} such pairs for which we have rough independence, which “ explains” the inequality in the proposition.

3.9 With this estimate, the end of the proof of the theorem is not difficult.

The sets L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) will always satisfy

t ∈ L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) ⇒ |t| ≤ 1 + |s| ≤ 1 + R.

We choose finite subsets Δ_{0} , Δ′_{0} , Δ_{2} of Σ_{nr}^{−}, Σ^{′−}, [−(1 + R + 2ρ), (1 + R + 2ρ)] respectively such that:

– Δ_{0} is ρ^{5/2} dense in Σ_{nr}^{−}, #Δ_{0} ≤ c_{8}ρ^{− \frac{5}{2} d};

– Δ′_{0} is ρ^{5/2} dense in Σ^{′−}, #Δ′_{0} ≤ c_{8}ρ^{− \frac{5}{2} d′};
– Δ_{2} is ρ^{5/2} dense in [−(1 + R + 2ρ), 1 + R + 2ρ], #Δ_{2} ≤ c_{8}ρ^{− \frac{5}{2}} .

Then, for each (\underline{θ}, \underline{θ}′) ∈ Δ_{0} × Δ′_{0} , we choose a finite subset Δ_{1}(\underline{θ}, \underline{θ}′) of the fiber of L̃ over (\underline{θ}, \underline{θ}′), which is ρ^{5/2} dense in this fiber, and has cardinality ≤ c_{8}ρ^{−5/2}. Let then

Δ = \{u = (\underline{θ}, \underline{θ}′, s, t), \underline{θ} ∈ Δ_{0}, \underline{θ}′ ∈ Δ′_{0}, t ∈ Δ_{2}, s ∈ Δ_{1}(\underline{θ}, \underline{θ}′)\}.

One has

#Δ ≤ c_{8}^{4} ρ^{−(5+ \frac{5}{2} (d+d′))} ≤ c_{8}^{4} ρ^{−10}.

Now, if ρ is small enough, we have

c_{8}^{4} ρ^{−10} exp ( − c_{7} ρ^{− \frac{1}{2k} (d+d′−1)}) < 1,

and therefore we can find \underline{ω}_{0} ∈ Ω such that, for any u ∈ Δ, we have either \underline{ω}_{0} ∉ Ω^{1}(u) or \underline{ω}_{0} ∈ Ω^{0}(u) (or both).

3.10 Fix \underline{ω}_{0} as above. Define L = L_{\underline{ω}_{0}} as follows:

L = \{(\underline{θ}, \underline{θ}′, s, t) ∈ L̃ × ℝ, ∃ (\underline{θ̃}, \underline{θ̃}′, s̃, t̃) ∈ L̃ × ℝ
with d(\underline{θ}, \underline{θ̃}) ≤ ρ^{5/2}, d(\underline{θ}′, \underline{θ̃}′) ≤ ρ^{5/2},
|s − s̃| ≤ ρ/2, |t − t̃| ≤ ρ/2
and t̃∈ L_{\underline{ω}_{0}}^{0}(\underline{θ̃}, \underline{θ̃}′, s̃)\}.

We claim that for (g^{\underline{ω}^{0}}, g′), L is a non empty compact recurrent set of relative configurations.

First, L contains L_{\underline{ω}_{0}}^{0} , hence is non empty. It is compact by construction. Let u = (\underline{θ}, \underline{θ}′, s, t) ∈ L.

We can find \underline{θ}^{0} ∈ Δ_{0} , \underline{θ}^{′0} ∈ Δ′_{0} , t_{0} ∈ Δ_{2} with d(\underline{θ}, \underline{θ}^{0}) < ρ^{5/2}, d(\underline{θ}′, \underline{θ}^{′0}) < ρ^{5/2}, |t − t_{0}| < ρ^{5/2}. Then, by construction of L̃, we have still (\underline{θ}^{0}, \underline{θ}^{′0}, s) ∈ L̃, hence we can find s_{0} ∈ Δ_{1}(\underline{θ}^{0}, \underline{θ}^{′0}) with |s − s_{0}| < ρ^{5/2}. Let u_{0} = (\underline{θ}^{0}, \underline{θ}^{′0}, s_{0}, t_{0}) ∈ Δ.

With \underline{θ̃}, \underline{θ̃}′, s̃, t̃ related to \underline{θ}, \underline{θ}′, s, t by the definition of L above, we have

d(\underline{θ}^{0}, \underline{θ̃}) < 2ρ^{5/2},
d(\underline{θ}^{′0}, \underline{θ̃}′) < 2ρ^{5/2},
|s_{0} − s̃| ≤ ρ/2 + ρ^{5/2} < ρ,
|t_{0} − t̃| ≤ ρ/2 + ρ^{5/2} < ρ,

which means that u_{0} ∈ L_{\underline{ω}_{0}}^{1} ; in other words, \underline{ω}_{0} ∈ Ω^{1}(u_{0}).

Then by choice of \underline{ω}_{0} , we have \underline{ω}_{0} ∈ Ω^{0}(u_{0}). This means that there exists \underline{b} ∈ Σ(ρ), \underline{b}′ ∈ Σ′(ρ), with b_{0} = θ_{0}^{0} , b′_{0} = θ_{0}^{′0} such that, if

T \underline{ω}_{0} ; \underline{b} T′_{\underline{b}′}(u_{0}) = (\underline{θ̂}^{0}, \underline{θ̂}^{′0}, ŝ_{0}, t̂_{0}) = û_{0} ,

we have

(θ̂^{0}, θ̂^{′0}, ŝ′_{0}) ∈ L̃ for |ŝ_{0} − ŝ′_{0}| < \frac{1}{2}ρ^{1/2}

and t̂_{0} ∈ L_{\underline{ω}_{0}}^{0}(\underline{θ̂}^{0}, \underline{θ̂}^{′0}, ŝ_{0}). Let

û = T \underline{ω}_{0} ; \underline{b}T′_{\underline{b}′}(u) = (\underline{θ̂}, \underline{θ̂}′, ŝ, t̂).

Lemma: We have

\begin{matrix}d(\underline{θ̂}, \underline{θ̂}^{0}) ≤ c_{9} ρ^{7/2}, & d(\underline{θ̂}′, \underline{θ̂}^{′0}) ≤ c_{9} ρ^{7/2} \\ |ŝ − ŝ_{0}| ≤ c_{9} ρ^{5/2}, & |t̂− t̂_{0}| ≤ c_{9} ρ^{3/2}. \end{matrix}

Proof: By proposition §1.3, the C^{1}-distance to the identity of k\underline{θ} ∘ (k^{\underline{θ}^{0}})^{−1}, k^{′\underline{θ}′} ∘ (k^{′\underline{θ}^{′0}})^{−1} is at most of order ρ^{5/2}. The estimates from the lemma then follow immediately from the definitions of T\underline{b}, T\underline{b}′ . □

To prove that û ∈ int L, we take (θ̂^{1}, θ̂^{′1}, ŝ_{1}, t̂_{1}) with

\begin{matrix}d(\underline{θ̂}^{1}, \underline{θ̂}) ≤ ρ^{7/2}, & d(\underline{θ̂}^{′1}, \underline{θ̂}′) ≤ ρ^{7/2} \\ |ŝ_{1} − ŝ| ≤ ρ^{5/2}, & |t̂_{1} − t̂| ≤ ρ^{3/2}. \end{matrix}

We have then

\begin{matrix}d(\underline{θ̂}^{1}, \underline{θ̂}^{0}) ≤ (1 + c_{9})ρ^{7/2}, & d(\underline{θ̂}^{′1}, \underline{θ̂}^{′0}) ≤ (1 + c_{9})ρ^{7/2}, \\ |ŝ_{1} − ŝ_{0}| ≤ (1 + c_{9})ρ^{5/2}, & |t̂_{1} − t̂_{0}| ≤ (1 + c_{9})ρ^{3/2} \end{matrix}

which imply first (\underline{θ̂}^{1}, \underline{θ̂}^{′1}, ŝ_{1}) ∈ L̃, and then (\underline{θ̂}^{1}, \underline{θ̂}^{′1}, ŝ_{1}, t̂_{1}) ∈ L.

3.11 In order to prove the theorem in §2.5, one has to modify slightly the argument as follows.

Let 0 < d̃ < d + d′ − 1; choose d̃ < d^{∗} < d + d′ − 1. We will indicate at the end of §4 how to define \overline{L}0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) (analogous to L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s)), for (\underline{θ}, \underline{θ}′, s) ∈ L̃.

For u ∈ L̃×ℝ, one defines \overline{Ω}^{1}(u) as Ω^{1}(u) above (using \overline{L}0 ; \underline{ω} instead of L0 ; \underline{ω}). Then, \overline{Ω}^{0}(u) is the set of \underline{ω} ∈ Ω such that, with M = [ρ^{−d^{∗}}], there exist pairs (\underline{b}^{1}, \underline{b}^{′1}), …, (\underline{b}^{M}, \underline{b}^{′M}) ∈ Σ(ρ) × Σ′(ρ) (with disjoint I(\underline{b}^{i})) for which properties (i), (ii) hold [we had M = 1 in §3.8].

We will see in section 5 that in this case we still have (as proposition 3.8)

ℙ(\overline{Ω}^{1}(u) − \overline{Ω}^{0}(u)) ≤ exp(−c̄_{7} ρ^{− \frac{1}{2k} (d+d′−1)}).

The set \overline{L}_{\underline{ω}_{0}} , defined as in 3.10 with the \overline{L}_{\underline{ω}_{0}}^{0} instead of L_{\underline{ω}_{0}}^{0} , is easily seen to be d̃-recurrent if \underline{ω}_{0} is selected as in 3.9.

3.12 Remark: Consider a smooth family (g_{μ}, g′_{μ}), μ ∈ [−1, +1]^{r} of maps as above such that one has HD(K_{μ}) + HD(K′_{μ}) > 1 for all μ.

We consider perturbations (g\underline{ω} ; μ, g′_{μ}) constructed as above, where now μ is restricted to a ρ^{3}-dense finite subset Δ_{3} of the parameter space [−1, 1]^{n}, with

# Δ_{3} ≤ c_{n} ρ^{−3n}.

As this is polynomial in ρ, we can select \underline{ω}_{0} such that, for any u ∈ Δ, μ ∈ Δ_{3} we have either \underline{ω}_{0} ∉ Ω_{μ}^{1}(u) or \underline{ω}_{0} ∈ Ω_{μ}^{0}(u).

For any μ_{0} ∈ Δ_{3} , this allows to construct (for (g_{μ_{0}}^{\underline{ω}_{0}}, g′_{μ_{0}})) a recurrent set L_{μ_{0}} .

But, in view of the construction of this set, it is fairly clear that L_{μ_{0}} is still recurrent for (g_{μ}^{\underline{ω}_{0}}, g′_{μ}), provided ||μ − μ_{0}|| < ρ^{3}.

Thus the following reinforcement of the theorem in §2.4 is true: there exists open and dense sets of families as above for which one can, for 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_{μ}, g′_{μ} 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 §2.4: the open set V ⊂ Ω_{Σ} × Ω_{Σ′} defined in §2.4 is not only dense in {(g, g′), HD(K) + HD(K′) > 1}, but its complement is infinite codimensional in the sense that generic nparameter families (for any n) take their values in V.

§4. Application of Marstrand’s theorem: construction of the sets E(\underline{a}, \underline{a}′) E(\underline{a}, \underline{a}′) E(\underline{a}, \underline{a}′) and

L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s).

4.1 Given a point (\underline{θ}, \underline{θ}′, s) in the space S of relative scales, and points x ∈ K(θ_{0}), x′ ∈ K′(θ′_{0}), there is a unique relative configuration above (\underline{θ}, \underline{θ}′, s) such that

A(k\underline{θ}(x)) = A′(k^{′\underline{θ}′}(x′)),

(where (\underline{θ}, A), (\underline{θ}′, A′) represents this configuration). Parametrizing the fiber of C over (\underline{θ}, \underline{θ}′, s) as in §3.3, the coordinate of this configuration is given by

t := π_{\underline{θ},\underline{θ}′,s}(x, x′)
= k^{′\underline{θ}′}(x′) − k^{′\underline{θ}′}(ω(θ′_{0})) − s(k\underline{θ}(x) − k\underline{θ}(ω(θ_{0}))).

4.2 We equip each set K(θ_{0}) (resp. K′(θ′_{0})) with the d-dimensional (resp. d′-dimensional) Hausdorff measure μ_{d} (resp. μ_{d′}).

Then, for (\underline{θ}, \underline{θ}′, s) ∈ S, we denote by μ(\underline{θ}, \underline{θ}′, s) the image under π_{\underline{θ},\underline{θ}′,s} of μ_{d} × μ_{d′} on K(θ_{0}) × K′(θ′_{0}).

It is well known that these exists constants c_{11} > c_{10} > 0 such that, for θ_{0} ∈ 𝔸, θ′_{0} ∈ 𝔸′:

c_{10} < μ_{d} × μ_{d′}(K(θ_{0}) × K′(θ′_{0})) < c_{11} .

Now, Kaufman’s proof of Marstrand’s theorem (see [PT2]) tells us that, for fixed (\underline{θ}, \underline{θ}′), the measure μ(\underline{θ}, \underline{θ}′, s) is absolutely continuous with respect to Lebesgue measure for Lebesgue almost every s, with L^{2}-density χ_{\underline{θ},\underline{θ}′,s} satisfying

∫_{J_{R}} ∥χ_{\underline{θ},\underline{θ}′,s}∥_{L^{2}}^{2} ds ≤ c_{12}(R),

where c_{12}(R) is independent on \underline{θ}, \underline{θ}′.

4.3 When one does control ∥χ_{\underline{θ},\underline{θ}′,s}∥_{L^{2}} , this gives, by Cauchy-Schwarz inequality, a lower bound for the Lebesgue measure of π_{\underline{θ},\underline{θ}′,s}(X), X being a subset of K × K′ with positive (d + d′)-dimensional Hausdorff measure; indeed we have:

μ_{d} × μ_{d′}(X) ≤ ∫_{π_{\underline{θ},\underline{θ}′,s}(X)} χ_{\underline{θ},\underline{θ}′,s}(t) dt
≤ |π_{\underline{θ},\underline{θ}′,s}(X)|^{1/2} ∥χ_{\underline{θ},\underline{θ}′,s}∥_{L^{2}}

and therefore

|π_{\underline{θ},\underline{θ}′,s}(X)| ≥ (μ_{d} × μ_{d′}(X))^{2} ∥χ_{\underline{θ},\underline{θ}′,s}∥_{L^{2}}^{−2} .

4.4 Fix (\underline{θ}, \underline{θ}′) in Σ^{−} × Σ^{′−}. Let \underline{a} ∈ Σ(ρ^{1/2k}), a′ ∈ Σ′(ρ^{1/2k}), with a_{0} = θ_{0} , a′_{0} = θ′_{0} . as seen in 3.1, one has

T\underline{a}T′_{\underline{a}′}(\underline{θ}, \underline{θ}′, s) = (\underline{θ}a_{1} … a_{n}, \underline{θ}′a_{1} … a_{n′} , εε′s|I\underline{θ}(\underline{a})| |I^{\underline{θ}′}(\underline{a}′)|^{−1})

where ε, ε′ ∈ {±1} and

c_{13}^{−1} ≤ |I\underline{θ}(\underline{a})| |I^{\underline{θ}′}(\underline{a}′)|^{−1} ≤ c_{13} .

We therefore have

∫_{J_{R}} ∥χ_{T\underline{a}T′_{\underline{a}′} (\underline{θ},\underline{θ}′,s)}∥_{L^{2}}^{2} ds ≤ c′_{12}(R),

with c′_{12}(R) independent of \underline{θ}, \underline{θ}′, \underline{a}, \underline{a}′. On the other hand, one has

# Σ(ρ^{1/2k}) ≤ c_{14} ρ^{−d/2k} ,
# Σ′(ρ^{1/2k}) ≤ c_{14} ρ^{−d′/2k} .

We conclude that

∫_{J_{R}} ∑_{\underline{a},\underline{a}′} ∥χ_{T\underline{a}T′_{\underline{a}′} (\underline{θ},\underline{θ}′,s)}∥_{L^{2}}^{2} ds ≤ c_{14}^{2} ρ^{− \frac{d+d′}{2k}} c′_{12}(R).

We now define, with c_{15} > 0 conveniently large to be determined later:

E(\underline{θ}, \underline{θ}′) = \{s ∈ J_{R} , ∥χ_{\underline{θ},\underline{θ}′,s}∥_{L^{2}}^{2} ≤ c_{15}

and ∑_{\underline{a},\underline{a}′} ∥χ_{T\underline{a}T′_{\underline{a}′} (\underline{θ},\underline{θ}′,s)}∥_{L^{2}}^{2} ≤ c_{15} ρ^{− \frac{d+d′}{2k}} \}.

For \underline{c} ∈ Σ(ρ^{1/2}), \underline{c}′ ∈ Σ′(ρ^{1/2}), we define E(\underline{c}, \underline{c}′) as the set of s ∈ J_{R} such that there exists \underline{θ}, \underline{θ}′ ending respectively with \underline{c}, \underline{c}′ such that s ∈ E(\underline{θ}, \underline{θ}′).

One has, for any \underline{θ} ∈ Σ^{−}, \underline{θ}′ ∈ Σ^{′−}:

m(J_{R} − E(\underline{θ}, \underline{θ}′)) ≤ 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(\underline{c}, \underline{c}′)) ≤ c_{1}

for all \underline{c} ∈ Σ(ρ^{1/2}), \underline{c}′ ∈ Σ′(ρ^{1/2}). This means that we can apply the Scale recurrence Lemma with the family E(\underline{c}, \underline{c}′) of subsets of J_{R}.

Let E^{∗}(\underline{c}, \underline{c}′) be then the family of subsets of J_{R} given by the Scale recurrence Lemma (cf. 3.7). We know that each E^{∗}(\underline{c}, \underline{c}′) is contained in a c_{2} ρ^{1/2}-neighbourhood of E(\underline{c}, \underline{c}′).

4.5 Until now, we have only worked with the unperturbed maps g, g′ (except from a possible initial perturbation, see §3.4). We now consider the family of random perturbations g\underline{ω} of §3.5, and will proceed to construct the sets L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s), for (\underline{θ}, \underline{θ}′, s) ∈ L̃. For \underline{a} ∈ Σ(ρ^{1/2k}), let Σ^{−}(\underline{a}) be the open and closed subset of Σ^{−} formed by the \underline{θ} ending with \underline{a}. Choose a subset Σ_{2}^{−} of Σ(ρ^{1/2k}) such that

Σ^{−} = ⋃_{Σ_{2}^{−}} Σ^{−}(\underline{a})

is a partition of Σ^{−}.

For \underline{a} ∈ Σ_{2}^{−}, define a subset Σ_{1}(\underline{a}) of the subset Σ_{1} of §3.5 (recall Σ_{1} ⊂ Σ(ρ^{1/k})), as the set of words in Σ_{1} starting with \underline{a}. For \underline{θ} ∈ Σ^{−}(\underline{a}), we also define Σ_{1}(\underline{θ}) = Σ_{1}(\underline{a}).

Let \underline{θ} ∈ Σ^{−}. We write

Ω = [−1, +1]^{Σ_{1}(\underline{θ})} × [−1, +1]^{Σ_{1}−Σ_{1}(\underline{θ})},
\underline{ω} = (\underline{ω}′, \underline{ω}′′)

and for such an \underline{ω}, we set

\underline{ω}^{∗} = (\underline{0}, \underline{ω}′′).

This depends on \underline{θ}, but nearby \underline{θ̂} (with d(\underline{θ}, \underline{θ̂}) < c_{0}^{−1} ρ^{1/2k}) will belong to the same Σ^{−}(\underline{a}) and give the same projection \underline{ω}^{∗} of \underline{ω}.

For (\underline{θ}, \underline{θ}′, s) ∈ L̃, the set L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) will actually only depend (as far as \underline{ω} is concerned) on the projection \underline{ω}^{∗} of \underline{ω} associated to \underline{θ}.

We will say that two words \underline{b}^{0}, \underline{b}^{1} ∈ Σ(ρ) are independent if there is no word \underline{b} ∈ Σ(ρ^{1/2k}) such that both \underline{b}^{0} and \underline{b}^{1} start with \underline{b}.

With c_{16} > 0 conveniently small, to be chosen in the following, let

N = [c_{16}^{2} ρ^{− \frac{1}{2k} (d+d′−1)}].

4.6 Let (\underline{θ}, \underline{θ}′, s) ∈ L̃, and \underline{ω} ∈ Ω.

We define L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) to be the set of points (\underline{θ}, \underline{θ}′, s, t) in the fiber for which there exist pairs (\underline{b}^{1}, \underline{b}^{′1}), …, (\underline{b}^{N}, \underline{b}^{′N}) in Σ(ρ) × Σ′(ρ), with b_{0}^{i} = θ_{0} , b_{0}^{′i} = θ′_{0} such that, if we set

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{b} & ^{i} \end{matrix} T′_{\underline{b}^{′i}}(\underline{θ}, \underline{θ}′, s, t) = (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}, t_{i}),

the following hold:

(i) the words \underline{b}^{1}, …, \underline{b}^{N} are pairwise independent; (ii) for 1 ≤ i ≤ N, \underline{θ}^{i} ∈ Σ_{nr}^{−};

(iii) for 1 ≤ i ≤ N, and |s̃ − s_{i}| ≤ \frac{2}{3}ρ^{1/2}, (\underline{θ}^{i}, \underline{θ}^{′i}, s̃) ∈ L̃;

(iv) for 1 ≤ i ≤ N, |t_{i}| ≤ 2(1 + R).

We will use also a slightly smaller set L−1 ; \underline{ω}(\underline{θ}, \underline{θ}′, s); it is defined in the same way than L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s), but with (iii), (iv) replaced by:

(iii)’ for 1 ≤ i ≤ N, and |s̃ − s_{i}| ≤ \frac{3}{4}ρ^{1/2}, (\underline{θ}^{i}, \underline{θ}^{′i}, s̃) ∈ L̃

(iv)’ for 1 ≤ i ≤ N, |t_{i}| ≤ 1 + R. In the end of this section, we will prove the following estimate.

Proposition: If c_{16} has been chosen sufficiently small, there exists c_{17} > 0 such that, for any (\underline{θ}, \underline{θ}′, s) ∈ L̃ and any \underline{ω} ∈ Ω, the Lebesgue measure of L−1 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) is > c_{17} .

4.7 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.

Lemma: Let \underline{θ} ∈ Σ^{−}, \underline{ω} ∈ Ω.

(i) for \underline{a} ∈ Σ^{+}, with a_{0} = θ_{0} , we have

|k^{\underline{θ},\underline{ω}}(h\underline{ω}(\underline{a})) − k\underline{θ}(h(\underline{a}))| ≤ c_{18}c_{5} ρ^{1− \frac{1}{2k}};

(ii) for \underline{a} a word with a_{0} = θ_{0} , |I(\underline{a})| > c_{0}^{−1} ρ, we have

|Log \frac{|I^{\underline{θ},\underline{ω}(\underline{a})|}}{|I^{\underline{θ}(\underline{a})}|} | ≤ c_{18}c_{5} ρ^{1− \frac{1}{2k}} .

The constant c_{18} is independent of \underline{θ}, \underline{ω}, \underline{a}, ρ, and the size c_{5} of the perturbation.

4.8 Let (\underline{θ}, \underline{θ}′, s) ∈ L̃ be given.

Choose a subfamily Σ_{2} of Σ(ρ^{1/2k}) of words starting with θ_{0} such that

K(θ_{0}) = ⋃_{Σ_{2}} K(\underline{a})

is a partition of K(θ_{0}). Similarly, choose a subfamily Σ′_{2} of Σ′(ρ^{1/2k}) of words starting with θ′_{0} such that

K′(θ′_{0}) = ⋃_{Σ′_{2}} K′(\underline{a}′).

For each (\underline{a}, \underline{a}′) ∈ Σ_{2} × Σ′_{2} , we have

c_{19}^{−1} ρ^{\frac{1}{2k} (d+d′)} ≤ μ_{d} × μ_{d′}(K(\underline{a}) × K′(\underline{a}′)) ≤ c_{19} ρ^{\frac{1}{2k} (d+d′)} .
Let J(\underline{a}, \underline{a}′) := π_{\underline{θ},\underline{θ}′,s}(I(\underline{a}) × I(\underline{a}′)) for \underline{a} ∈ Σ_{2} , \underline{a}′ ∈ Σ_{2} ; we have
c_{19}^{−1} ρ^{\frac{1}{2k}} ≤ |J(a, a′)| ≤ c_{19} ρ^{\frac{1}{2k}} .

Say (\underline{a}, \underline{a}′) is good if there are no more than c_{16}^{−1} ρ^{−1/2k(d+d′−1)} intervals J(\underline{ã}, \underline{ã}′) whose centers are distant from the center of J(\underline{a}, \underline{a}′) by less than 2c_{19} ρ^{1/2k}.

Lemma: The number of bad pairs (\underline{a}, \underline{a}′) is less than 12c_{19}^{3} c_{16}c_{15} ρ^{−1/2k(d+d′)}.

Proof of lemma: By construction of L̃, there exists \underline{θ̃}, \underline{θ̃}′, s̃ with d(\underline{θ}, \underline{θ̃}) ≤ (c_{2} + c_{0})ρ^{1/2}, d(\underline{θ}′, \underline{θ̃}′) ≤ (c_{2} + c_{0})ρ^{1/2}, |s − s̃| ≤ c_{2} ρ^{1/2} such that

∥χ_{\underline{θ̃},\underline{θ̃}′,s̃}∥_{L^{2}}^{2} ≤ c_{15} .

Therefore, if we replace \underline{θ}, \underline{θ}′, s by \underline{θ̃}, \underline{θ̃}′, s̃ the endpoints of J(\underline{a}, \underline{a}′) only move by a distance of order at most ρ^{1/2}. It is thus sufficient to prove the lemma for \underline{θ̃}, \underline{θ̃}′, s̃ with 2c_{19} ρ^{1/2k} replaced by 5/2 c_{19} ρ^{1/2k}.

Let (\underline{a}, \underline{a}′) be bad; then there are at least c_{16}^{−1} ρ^{−1/2k(d+d′−1)} pairs (ã, ã′) such that J(ã, ã′) is contained in the interval J^{∗}(\underline{a}, \underline{a}′) of same center than J(\underline{a}, \underline{a}′) and length 6 c_{19} ρ^{1/2k}. This means that

∫_{J^{∗}(\underline{a},\underline{a}′)} χ_{\underline{θ̃},\underline{θ̃}′,s̃} ≥ c_{19}^{−1} c_{16}^{−1} ρ^{1/2k} = \frac{1}{6}c_{19}^{−2} c_{16}^{−1}|J^{∗}(\underline{a}, \underline{a}′)|.

But then, by Cauchy-Schwarz

\frac{1}{6}c_{19}^{−2} c_{16}^{−1}|J^{∗}(\underline{a}, \underline{a}′)| ∫_{J^{∗}(\underline{a},\underline{a}′)} χ_{\underline{θ̃},\underline{θ̃}′,s̃} ≤ ( ∫_{J^{∗}(\underline{a},\underline{a}′)} χ_{\underline{θ̃},\underline{θ̃}′,s̃})^{2}
≤ |J^{∗}(a, a′)| ∫_{J^{∗}(a,a′)} χ_{\underline{θ̃},\underline{θ̃}′,s̃}^{2}

and thus

∫_{J^{∗}(\underline{a},\underline{a}′)} χ_{\underline{θ̃},\underline{θ̃}′,s̃}^{2} ≥ \frac{1}{6}c_{19}^{−2} c_{16}^{−1} ∫_{J^{∗}(\underline{a},\underline{a}′)} χ_{\underline{θ̃},\underline{θ̃}′,s̃}

Let J^{∗} be the union over all bad pairs (\underline{a}, \underline{a}′) of the intervals J^{∗}(\underline{a}, \underline{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

∫_{J^{∗}} χ_{\underline{θ̃},\underline{θ̃}′,s̃}^{2} ≥ \frac{1}{12}c_{19}^{−2} c_{16}^{−1} ∫_{J^{∗}} χ_{\underline{θ̃},\underline{θ̃}′,s̃}.

As ∥χ_{\underline{θ̃},\underline{θ̃}′,s̃}∥_{L^{2}}^{2} ≤ c_{15} , we get

∫_{J^{∗}} χ_{\underline{θ̃},\underline{θ̃}′,s̃} ≤ 12 c_{19}^{2} c_{16} c_{15} .

As J^{∗} contains J(\underline{a}, \underline{a}′) for all bad (\underline{a}, \underline{a}′), the estimate of the lemma follows. □ 4.9 To keep the argument clearer, we now prove Proposition 4.6 when \underline{ω} = 0 (for the unperturbed map g). Afterwards we will indicate how to modify the argument for general \underline{ω}.

First, we construct the pairs (\underline{b}, \underline{b}′) amongst which the pairs (\underline{b}^{i}, \underline{b}^{′i}) of 4.6 must be looked for. We make the following easy observation: Lemma: Let \underline{θ} ∈ Σ_{nr}^{−}. The number of words \underline{c} ∈ Σ(ρ^{1/2}) with c_{0} = θ_{0} , such that \underline{θ}c_{1} … c_{m} ∉ Σ_{nr}^{−} is o(ρ^{−1/2d}) as ρ → 0, uniformly in \underline{θ}.

Now let (\underline{θ}, \underline{θ}′, s) ∈ L̃. It follows from conclusion (ii) of the Scale recurrence Lemma and the last observation that we can find at least \frac{1}{2}c_{3} ρ^{−1/2(d+d′)} pairs (\underline{c}^{i}, \underline{c}^{′i}) ∈ Σ(ρ^{1/2})×Σ′(ρ^{1/2}) such that, writing T\underline{c}^{i}T′_{\underline{c}^{′i}}(\underline{θ}, \underline{θ}′, s) = (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}), we have:

– \underline{θ}^{i} ∈ Σ_{nr}^{−};

– [s_{i} − ρ^{1/2}, s_{i} + ρ^{1/2}] ⊂ E^{∗}(\underline{c}^{i}, \underline{c}^{′i}).

As (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}) again belongs to L̃, we can for each i find at least \frac{1}{2}c_{3} ρ^{−1/2(d+d′)} pairs (\underline{d}^{ij}, \underline{d}^{′ij}) ∈ Σ(ρ^{1/2}) × Σ′(ρ^{1/2}) (with the first letter of \underline{d}^{ij}, \underline{d}^{′ij} being the last one of \underline{c}^{i}, \underline{c}^{′i} respectively), such that writing T\underline{d}^{ij}T′_{\underline{d}′_{ij}}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}) = (\underline{θ}^{ij}, \underline{θ}^{′ij}, s_{ij}), we have

– \underline{θ}^{ij} ∈ Σ_{nr}^{−};

– [s_{ij} − ρ^{1/2}, s_{ij} + ρ^{1/2}] ∈ E^{∗}(\underline{d}^{ij}, \underline{d}^{′ij}).

Concatenation of the \underline{c}^{i}, \underline{c}^{′i} and \underline{d}^{ij}, \underline{d}^{′ij} gives a family of words (\underline{b}^{ij}, \underline{b}^{′ij}) in Σ(ρ) × Σ′(ρ) with at least \frac{1}{4}c_{3}^{2} ρ^{−(d+d′)} elements.

Let (\underline{θ̃}, \underline{θ̃}′, s̃) be such that d(\underline{θ}, θ̃) ≤ (c_{0}+c_{2})ρ^{1/2}, d(\underline{θ}′, \underline{θ̃}′) ≤ (c_{0}+c_{2})ρ^{1/2}, |s−s̃| ≤ c_{2} ρ^{1/2} and s̃ ∈ E(θ̃, θ̃′). Lemma: If c_{16} has been chosen sufficiently small, there are at least \frac{1}{6}c_{3} c_{19}^{−2} ρ^{− \frac{d+d′}{2k}} pairs (\underline{a}, \underline{a}′) ∈ Σ_{2} × Σ′_{2} which are good and satisfy

∥χ_{T\underline{a}T′_{\underline{a}′} (θ̃,θ̃′,s̃)}∥_{L^{2}}^{2} ≤ c_{16}^{−1}

and such that at least \frac{1}{6}c_{3}c_{19}^{−1} ρ^{−(d+d′)( \frac{1}{2} − \frac{1}{2k})} pairs (\underline{c}^{i}, \underline{c}^{′i}) start with (\underline{a}, \underline{a}′).

Proof: By definition of E(\underline{θ̃}, \underline{θ̃}′), the number of pairs (\underline{a}, \underline{a}′) ∈ Σ_{2} × Σ′_{2} such that

∥χ_{T\underline{a}T′_{\underline{a}′} (θ̃,θ̃′,s̃)}∥_{L^{2}}^{2} > c_{16}^{−1}

is at most c_{16}c_{15} ρ^{− \frac{1}{2k} (d+d′)}. Call (\underline{a}, \underline{a}′) very good if it is good and satisfy

∥χ_{T\underline{a}T′_{\underline{a}′} (θ̃,θ̃′,s̃)}∥_{L^{2}}^{2} ≤ c_{16}^{−1}

The number of pairs (\underline{a}, \underline{a}′) which are not very good is thus ≤ c_{15}c_{16}(1 + 12c_{19}^{3})ρ^{− \frac{1}{2k} (d+d′)}.

But for each (\underline{a}, \underline{a}′) ∈ Σ_{2} ×Σ′_{2} , the number of pairs (\underline{c}, \underline{c}′) ∈ Σ(ρ^{1/2})×Σ′(ρ^{1/2}) starting

with (\underline{a}, \underline{a}′) is at most c_{19}^{2} ρ^{(d+d′)( \frac{1}{2k} − \frac{1}{2} )}.

It follows, if c_{16} is small enough, that at least \frac{1}{3}c_{3} ρ^{−1/2(d+d′)} pairs (\underline{c}^{i}, \underline{c}^{′i}) start with

a very good (\underline{a}, \underline{a}′) ∈ Σ_{2} × Σ′_{2} .

The total number of pairs (\underline{a}, \underline{a}′) is at most c_{19} ρ^{− \frac{d+d′}{2k}}; if we discard those pairs for

which less than \frac{1}{6}c_{3}c_{19}^{−1} ρ^{−(d+d′)( \frac{1}{2} − \frac{1}{2k})} pairs (\underline{c}^{i}, \underline{c}^{′i}) begin with (\underline{a}, \underline{a}′), we are still left with

\frac{1}{6}c_{3} ρ^{−1/2(d+d′)} pairs (\underline{c}^{i}, \underline{c}^{′i}), of which no more than c_{19}^{2} ρ^{−(d+d′)( \frac{1}{2} − \frac{1}{2k})} may begin with the

same (\underline{a}, \underline{a}′). □

4.10 Let us call excellent those pairs (\underline{a}, \underline{a}′) given by the last lemma.

Recall that we have defined

J(\underline{a}, \underline{a}′) = π_{\underline{θ},\underline{θ}′,s}(I(\underline{a}) × I(\underline{a}′)),

and define similarly J(\underline{c}^{i}, \underline{c}^{′i}), J(\underline{b}^{ij}, \underline{b}^{′ij}). We only consider those i for which (\underline{c}^{i}, \underline{c}^{′i}) starts with an excellent (\underline{a}, \underline{a}′).

The following general lemmas will be used repeteadly:

Lemma 1: Let (J_{α})_{α∈A}, (J′_{α})_{α∈A} be two finite families of intervals, and let λ > 1, ε > 0, ν > 0.

Assume that for every α, we have:

(i) ε < |J_{α}| < λε, ε < |J′_{α}|; (ii) the centers of J_{α}, J′_{α} are distant by no more than νε.

Then

m( ⋃_{A} J′_{α}) ≥ \frac{1}{λ(4ν + 4)}m( ⋃_{A} J_{α}).

Proof: Let A′ ⊂ A be a minimal family with

⋃_{A′} J_{α} = ⋃_{A} J_{α} .

Then each point in ⋃_{A} J_{α} belongs at most to two intervals J_{α}, α ∈ A′. Replace each J′_{α},

α ∈ A′, by the interval J̃′_{α} with same center and length ε. One has

∑_{A′} |J̃′_{α}| ≥ λ^{−1} ∑_{A′} |J_{α}| ≥ λ^{−1}| ⋃_{A} J_{α}|.

On the other hand, let x ∈ ℝ; if x ∈ J̃′_{α}, for α ∈ A′, then the center of J̃′_{α} belongs to [x− \frac{ε}{2}, x+ \frac{ε}{2}] and the center of J_{α} is contained in [x−(ν+ \frac{1}{2})ε, x+(ν+ \frac{1}{2})ε]. Replacing the J_{α}

by intervals J̃_{α} with same center and length ε, we see that there are at most (4ν +4) α ∈ A′ such that x ∈ J̃′_{α}. Then we have:

|⋃_{A} J′_{α}| ≥ |⋃_{A′} J̃′_{α}| ≥ \frac{1}{4ν + 4}Σ|J̃′_{α}|. □

Lemma 2: Let (J_{α})_{α∈A} be a finite family of intervals, and λ > 1. For each α, let K_{α} be a subset of J_{α} with m(K_{α}) ≥ λ^{−1}|J_{α}|. Then

m( ⋃_{A} K_{α}) ≥ \frac{λ^{−1}}{2} m( ⋃_{A} J_{α}).

Proof: We extract a subfamily A′ such that ⋃_{A′} J_{α} = ⋃_{A} J_{α} and each x belongs to at most two J_{α}, α ∈ A′. Then we have

m( ⋃_{A} K_{α}) ≥ m( ⋃_{A′} K_{α}) ≥ \frac{1}{2} ∑_{A′} m(K_{α})

≥ \frac{λ^{−1}}{2} ∑_{A′} |J_{α}|. □

4.11 Now let (\underline{a}, \underline{a}′) be an excellent pair. For i such that (\underline{c}^{i}, \underline{c}^{′i}) starts with (\underline{a}, \underline{a}′), let J̃(\underline{c}^{i}, \underline{c}^{′i}) = π_{\underline{θ̃},\underline{θ̃}′,s̃}(I(\underline{c}^{i})×I(\underline{c}^{′i})), and let J̃_{1}(\underline{a}, \underline{a}′) be the union of those J̃(\underline{c}^{i}, \underline{c}^{′i}). The μ_{d}×μ_{d′}- measure of the union of the K(\underline{c}^{i}) × K′(\underline{c}^{′i}) is ≥ c_{20} ρ^{− \frac{1}{2k} (d+d′)} (because (\underline{a}, \underline{a}′) is excellent), and follows then from 4.3 that

|J̃_{1}(\underline{a}, \underline{a}′)| ≥ c_{21} c_{16} ρ^{−1/2k}.

When we come back to (\underline{θ}, \underline{θ}′, s) and define J_{1}(\underline{a}, \underline{a}′) to be the union of the J(\underline{c}^{i}, \underline{c}^{′i}) starting with (\underline{a}, \underline{a}′), the sizes of the J(\underline{c}^{i}, \underline{c}^{′i}) are still of order ρ^{1/2} and the centers are distant from those of J̃(\underline{c}^{i}, \underline{c}^{′i}) by distances of order at most ρ^{1/2}. Then, by Lemma 1 above, we get

|J_{1}(\underline{a}, \underline{a}′)| ≥ c_{22}^{−1} c_{16} ρ^{1/2k}.

Similarly, for each (\underline{c}^{i}, \underline{c}^{′i}) (starting with (\underline{a}, \underline{a}′)), let J_{1}(\underline{c}^{i}, \underline{c}^{′i}) be the union of the

J(\underline{b}^{ij}, \underline{b}^{′ij}) starting with (\underline{c}^{i}, \underline{c}^{′i}). Then are at least \frac{1}{2}c_{3} ρ^{−1/2(d+d′)} such (\underline{b}^{ij}, \underline{b}^{′ij}), and (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}) =

T\underline{c}^{i}T′_{\underline{c}^{′i}}(\underline{θ}, \underline{θ}′, s) belongs to L̃.

Therefore, using again 4.3, we conclude that

|J_{1}(\underline{c}^{i}, \underline{c}^{′i})| ≥ c_{23}^{−1}|J(\underline{c}^{i}, \underline{c}^{′i})|.

[The argument is the same than above: we first consider (\underline{θ̃}^{i}, \underline{θ̃}^{′i}, s̃_{i}), with d(\underline{θ}^{i}, \underline{θ̃}^{i}), d(\underline{θ}^{′i}, \underline{θ̃}^{′i}), |s_{i} − s̃_{i}| of order ρ^{1/2}, such that ∥χ_{\underline{θ̃}^{i},\underline{θ̃}^{′i},s̃_{i}} ∥_{L^{2}}^{2} ≤ c_{15}].

Finally, let J_{2}(\underline{a}, \underline{a}′) be the union over those (\underline{c}^{i}, \underline{c}^{′i}) starting with (\underline{a}, \underline{a}′) of the sets J_{1}(\underline{c}^{i}, \underline{c}^{′i}). By Lemma 2 above, we obtain that

|J_{2}(\underline{a}, \underline{a}′)| ≥ \frac{1}{2}c_{23}^{−1} c_{22}^{−1} c_{16} ρ^{1/2k}.

Let φ (resp. φ_{2}) be the sum, over excellent pairs (\underline{a}, \underline{a}′), of the characteristic functions of J(\underline{a}, \underline{a}′) (resp. J_{2}(\underline{a}, \underline{a}′)).

The number of excellent pairs is ≥ \frac{1}{6}c_{3}c_{19}^{−2} ρ^{− \frac{d+d′}{2k}} by 4.8, and therefore

∫ φ_{2} ≥ \frac{1}{12}c_{23}^{−1} c_{22}^{−1} c_{3} c_{19}^{2} c_{16} ρ^{− \frac{1}{2k} (d+d′−1)}.

On the other hands, because excellent pairs are good, one has

φ_{2} ≤ φ ≤ c_{16}^{−1} ρ^{−1/2k(d+d′−1)}.

We conclude that

m(\{φ_{2} ≥ c_{16}^{2} ρ^{− \frac{1}{2k} (d+d′−1)}\}) ≥ c_{24} c_{16}^{2} := c_{17}

if c_{16} > 0 is small enough (recall that the support of φ_{2} is contained in an interval of length 1 + R).

But, if φ_{2}(t) ≥ c_{16}^{2} ρ^{− \frac{1}{2k} (d+d′−1)}, this means that there are at least N excellent pairs (\underline{a}^{1}, \underline{a}^{′1}), …, (\underline{a}^{N}, \underline{a}^{′N}), and for each of these pairs (\underline{a}^{ℓ}, \underline{a}^{′ℓ}), an element (\underline{b}^{ij}, \underline{b}^{′ij}) starting with (\underline{a}^{ℓ}, \underline{a}^{′ℓ}), such that t ∈ J(\underline{b}^{ij}, \underline{b}^{′ij}); renaming (\underline{b}^{ij}, \underline{b}^{′ij}) = (\underline{b}^{ℓ}, \underline{b}^{′ℓ}), and setting

T\underline{b}_{ℓ}T′_{\underline{b}^{′ℓ}}(\underline{θ}, \underline{θ}′, s, t) = (\underline{θ}^{ℓ}, \underline{θ}^{′ℓ}, s_{ℓ}, t_{ℓ}),

conditions (i), (ii) of 4.6 are satisfied; one has by the choice of the \underline{b}^{ij}, \underline{b}^{′ij}: (iii)” (\underline{θ}^{ℓ}, \underline{θ}^{′ℓ}, s̃) ∈ L̃ if |s̃ − s_{ℓ}| ≤ ρ^{1/2}; finally, t ∈ J(\underline{b}^{ℓ}, \underline{b}^{′ℓ}) ⇒ |t_{ℓ}| ≤ 1 + R.

Hence (i), (ii), (iii)”, (iv)’ are satisfied and t ∈ L−1 ; \underline{0}(\underline{θ}, \underline{θ}′, s). We have proved that

|L−1 ; \underline{0}(\underline{θ}, \underline{θ}′, s)| ≥ c_{17} .

4.12 We now consider the random perturbation g\underline{ω} instead of g. By definition, we have to work with g^{\underline{ω}^{∗}}.

By 4.7, doing this, the endpoints of each J(\underline{a}, \underline{a}′) are moved by no more than c_{18}c_{5} ρ^{1− \frac{1}{2k}}. So if ρ is small enough, and (\underline{a}, \underline{a}′) is good, there will be (for g^{\underline{ω}^{∗}}) no more than c_{16}^{−1} ρ^{− \frac{1}{2k} (d+d′−1)} intervals J(ã, ã′) whose centers are distant from the center of J(a, a′) by less than \frac{7}{4}c_{19} ρ^{\frac{1}{2k}} [instead of 2c_{19} ρ^{1/2k} in the unperturbed situation].

When we consider the pairs (\underline{c}^{i}, \underline{c}^{′i}), (\underline{d}^{ij}, \underline{d}^{′ij}), (\underline{b}^{ij}, \underline{b}^{′ij}) of the beginning of 4.8, the conditions \underline{θ}^{i} ∈ Σ_{nr}^{−}, \underline{θ}^{ij} ∈ Σ_{nr}^{−} are independent of \underline{ω}. If we write

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{c} & _{i} \end{matrix}T′_{\underline{c}′_{i}}(\underline{θ}, \underline{θ}′, s) = (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}^{∗})),

we have |s_{i}(\underline{ω^{∗}}) − s_{i}| ≤ c_{5}c′_{18} ρ^{1− \frac{1}{2k}} by Lemma 4.7 (ii); writing

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{d} & _{ij} \end{matrix}T′_{\underline{d}′_{ij}}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}) = (\underline{θ}^{ij}, \underline{θ}^{′ij}, s_{ij}^{0}(\underline{ω}^{∗})),

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{d} & _{ij} \end{matrix}T′_{\underline{d}′_{ij}}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}^{∗}) = (\underline{θ}^{ij}, \underline{θ}^{′ij}, s_{ij}(\underline{ω}^{∗})),

we have

|s_{ij}(\underline{ω}^{∗}) − s_{ij}^{0}(\underline{ω}^{∗})| ≤ c_{5}c′′_{18} ρ^{1− \frac{1}{2k}} ,
|s_{ij}^{0}(\underline{ω}^{∗}) − s_{ij}| ≤ c_{5}c′_{18} ρ^{1− \frac{1}{2k}} ,

and thus

|s_{ij} − s_{ij}(\underline{ω}^{∗})| ≤ c_{5}(c′_{18} + c′′_{18})ρ^{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^{\underline{ω}^{∗}} and excellent pair (\underline{a}, \underline{a}′) the subsets J_{1}^{\underline{ω}^{∗}}(\underline{a}, \underline{a}′), J_{2}^{\underline{ω}^{∗}}(\underline{a}, \underline{a}′) of J^{\underline{ω}^{∗}}(\underline{a}, \underline{a}′).

By the estimates of Lemma 4.7 and the Lemma 1 of 4.10, we will still have

|J_{1}^{\underline{ω}^{∗}}(\underline{a}, \underline{a}′)| ≥ c_{22}^{′−1} c_{16} ρ^{\frac{1}{2k}} .

For the same reasons, we will get

|J_{1}^{\underline{ω}^{∗}}(\underline{c}^{i}, \underline{c}^{′i})| ≥ c_{23}^{′−1}|J^{\underline{ω}^{∗}}(\underline{c}^{i}, \underline{c}^{′i})|

and this allows to prove in exactly the same way that

m(\{φ_{2}^{\underline{ω}^{∗}} ≥ c_{16}^{2} ρ^{− \frac{1}{2k} (d+d′−1)}\}) ≥ c′_{17} .

The proof of

φ_{2}^{\underline{ω}^{∗}}(t) ≥ c_{16}^{2} ρ^{− \frac{1}{2k} (d+d′−1)} ⇒ t ∈ L−1 ; \underline{ω}(\underline{θ}, \underline{θ}′, s)

is the same, except that we have

|s_{ℓ}(\underline{ω}^{∗}) − s_{ℓ}| ≤ c_{5}(c′_{18} + c′′_{18})ρ^{1− \frac{1}{2k}} ,

which gives (iii)’ instead of (iii)”.

4.13 If we want to prove the reinforcement of theorem 2.4 given in 2.5, one has to modify the definition of L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) (for (\underline{θ}, \underline{θ}′, s) ∈ L̃, \underline{ω} ∈ Ω) as follows.

Let d̃< d + d′ − 1; choose d̃< d^{∗} < d + d′ − 1, and k large enough to have

(1 − \frac{1}{k} )(d + d′ − 1) > d^{∗}.

Let M = [ρ^{−d^{∗}}]. We define \overline{L}0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) to be set of points (\underline{θ}, \underline{θ}′, s, t) in the fiber for which there exist pairs (\underline{b}_{m}^{ℓ}, \underline{b}_{m}^{′ℓ}), 1 ≤ ℓ ≤ N, 1 ≤ m ≤ M such that conditions (ii), (iii), (iv) of §4.6 are satisfied for all ℓ, m and moreover (i’) for any 1 ≤ m ≤ M, the words \underline{b}_{m}^{1}, …, \underline{b}_{m}^{N} are independent.

We also ask that

(v) I(\underline{b}_{m}^{ℓ}) ∩ I(b_{\underline{m}′}^{ℓ′}) = ϕ if (ℓ, m) ≠ (ℓ′, m′). Replacing (iii), (iv) by (ii)’, (iv)’ we also define \overline{L}−1 ; \underline{ω}(\underline{θ}, \underline{θ}′, s). We now claim that proposition 4.6 still holds:

|\overline{L}−1 ; \underline{ω}(\underline{θ}, \underline{θ}′, s)| > c̄_{17} .

To prove this, we proceed as above: for an excellent pair (\underline{a}, \underline{a}′) and (\underline{c}^{i}, \underline{c}^{′i}) starting with (\underline{a}, \underline{a}′), we now define \overline{J}_{1}(\underline{c}^{i}, \underline{c}^{′i}) as the set of t which belong to at least ρ^{− \frac{1}{2} d^{∗}} intervals J(\underline{b}^{ij}, \underline{b}^{′ij}) (with (\underline{b}^{ij}, \underline{b}^{′ij}) starting with (\underline{c}^{i}, \underline{c}^{′i})). As \frac{1}{2} d^{∗} < \frac{1}{2} (d + d′ − 1), one easily sees that we still have

|\overline{J}_{1}(\underline{c}^{i}, \underline{c}^{′i})| ≥ c̄_{23}^{−1}|J(\underline{c}^{i}, \underline{c}^{′i})|.

Next we define \overline{J}_{2}(\underline{a}, \underline{a}′) as the set of t which belong to at least ρ^{−1/2 d^{∗}} sets \overline{J}_{1}(\underline{c}^{i}, \underline{c}^{′i})

(with (\underline{c}^{i}, \underline{c}^{′i}) starting with (\underline{a}, \underline{a}′)). Because \frac{1}{2} d^{∗} < (\frac{1}{2} − \frac{1}{2k} )(d + d′ − 1), we will still have

|\overline{J}_{2}(\underline{a}, \underline{a}′)| ≥ c̄_{23}^{−1} ρ^{1/2k}.

If φ̄_{2} is the sum, over excellent pairs, of the characteristic functions of the \overline{J}_{2}(\underline{a}, \underline{a}′), the same argument than for φ_{2} shows that

m({φ̄_{2} ≥ N}) ≥ c̄_{17} .

If φ̄_{2}(t) ≥ N, it is now easy to construct the (\underline{b}_{m}^{ℓ}, \underline{b}_{m}^{′ℓ}) as required: the index ℓ refers to the excellent pairs such that t ∈ \overline{J}_{2}(\underline{a}, \underline{a}′), and the index M ≤ ρ^{− \frac{1}{2} d^{∗}} · ρ^{− \frac{1}{2} d^{∗}} for the possibilities for fixed (\underline{a}, \underline{a}′).

§5. The family of random perturbations.

5.1 In this section, we will prove the estimates of lemma §4.7 and the proposition §3.8. We begin with lemma §4.7, part (i).

We are given \underline{θ} ∈ Σ^{−} and \underline{a} ∈ Σ^{+}, with a_{0} = θ_{0} . Let \underline{ω} ∈ Ω; we want to show that

|k^{\underline{θ},\underline{ω}}(h\underline{ω}(\underline{a})) − k\underline{θ}(h(\underline{a}))| ≤ c_{18} c_{5} ρ^{1− \frac{1}{2k}} .

Let â = (a_{0}, …, a_{m}) be an initial part of \underline{a} which belongs to Σ(ρ^{1−1/2k}). The sizes of I\underline{θ}(â) and I^{\underline{θ},\underline{ω}}(â) have order ρ^{1−1/2k}, and we have to show that these two intervals are distant by no more than c_{5}c_{18} ρ^{1−1/2k}.

To simplify notation, we just write f̃_{aa′} for the inverse branches associated to g\underline{ω}. Because the C^{0}-distance between f_{aa′} and f̃_{aa′} is c_{5} ρ^{1+1/2k}, and these maps are con-

tractions, the endpoints of I\underline{ω}(â) are distant from those of I(â) by no more than a multiple of c_{5} ρ^{1+1/2k}. So it is sufficient to see that the C^{0}-distance between k\underline{θ} and k^{\underline{θ},\underline{ω}} := k̃\underline{θ} is no more than c_{18}c_{5} ρ^{1−1/2k}.

5.2 Let z ∈ I(θ_{0}); set

x_{0} = x̃_{0} = 0, y_{0} = ỹ_{0} = 1, z_{0} = z̃_{0} = z,

and define by induction

x_{n} = f_{θ_{−n},θ_{−n+1}}(x_{n−1}), y_{n} = f_{θ_{−n},θ_{−n+1}}(y_{n−1})
z_{n} = f_{θ_{−n},θ_{−n+1}}(z_{n−1}), x̃_{n} = f̃_{θ_{−n},θ_{−n+1}}(x̃_{n−1}),
ỹ_{n} = f̃_{θ_{−n},θ_{−n+1}}(ỹ_{n−1}), z̃_{n} = f̃_{θ_{−n},θ_{−n+1}}(z̃_{n−1}).

We will just write f_{n}, f̃_{n} for f_{θ_{−n},θ_{−n+1}}, f̃_{θ_{−n},θ_{−n+1}} . We have

k\underline{θ}(z) = Lim_{n→+∞} \frac{z_{n} − x_{n}}{y_{n} − x_{n}} ,

k̃\underline{θ}(z) = Lim_{n→+∞} \frac{z̃_{n} − x̃_{n}}{ỹ_{n} − x̃_{n}} ·

Because f_{n}, f̃_{n} are C^{0}-distant by c_{5} ρ^{1+1/2k}, we will always have |z_{n} − z̃_{n}| ≤ cc_{5} ρ^{1+1/2k},

|x_{n} − x̃_{n}| ≤ cc_{5} ρ^{1+1/2k},
|y_{n} − ỹ_{n}| ≤ cc_{5} ρ^{1+1/2k}.

Thus, as long as |y_{n} − x_{n}| ≥ c_{0}^{−3} ρ^{−1/k}, one has

|\frac{z_{n} − x_{n}}{y_{n} − x_{n}} − \frac{z̃_{n} − x̃_{n}}{ỹ_{n} − x̃_{n}} | ≤ cc_{5} ρ^{1− \frac{1}{2k}} .

Let n_{0} be the largest such integer; we will have |y_{n} − x_{n}| ≤ c_{0}^{−2} ρ^{−1/k} for n ≥ n_{0} . Therefore on an interval containing the six points x_{n}, y_{n}, z_{n}, x̃_{n}, ỹ_{n}, z̃_{n}, we will have (cf. 3.6.3) either

f̃_{n+1}(u) = f_{n+1}(u)

or

f̃_{n+1}(u) = f_{n+1}(u) + c_{5} ω_{n+1} ρ^{1+1/2k}

(where ω_{n+1} is a component of \underline{ω}). Thus we will have, for n ≥ n_{0} :

z_{n+1} − x_{n+1} = (z_{n} − x_{n}) ∫_{0}^{1} Df_{n+1}(x_{n} + t(z_{n} − x_{n})) dt,
z̃_{n+1} − x̃_{n+1} = (z̃_{n} − x̃_{n}) ∫_{0}^{1} Df_{n+1}(x̃_{n} + t(z̃_{n} − x̃_{n})) dt,

and similarly for y_{n+1} − x_{n+1} , ỹ_{n+1} − x̃_{n+1} .

Write

\frac{z_{n+1} − x_{n+1}}{y_{n+1} − x_{n+1}} = \frac{z_{n} − x_{n}}{y_{n} − x_{n}} α_{n} ,

\frac{z̃_{n+1} − x̃_{n+1}}{ỹ_{n+1} − x̃_{n+1}} = \frac{z̃_{n} − x̃_{n}}{ỹ_{n} − x̃_{n}} α̃_{n} ,

with

α_{n} = \frac{∫_{0}^{1} Df_{n+1}(x_{n} + t(z_{n} − x_{n}))dt}{∫_{0}^{1} Df_{n+1}(x_{n} + t(y_{n} − x_{n}))dt} ,
α̃_{n} = \frac{∫_{0}^{1} Df_{n+1}(x̃_{n} + t(z̃_{n} − x̃_{n}))dt}{∫_{0}^{1} Df_{n+1}(x̃_{n} + t(ỹ_{n} − x̃_{n}))dt} ·

We have

|α_{n} − 1| ≤ c|y_{n} − x_{n}|, |α̃_{n} − 1| ≤ c|ỹ_{n} − x̃_{n}|

and also

|α_{n} − α̃_{n}| ≤ cc_{5} ρ^{1+1/2k}.

We obtain, setting

\frac{z̃_{n+1} − x̃_{n+1}}{ỹ_{n+1} − x̃_{n+1}} − \frac{z_{n+1} − x_{n+1}}{y_{n+1} − x_{n+1}} = w_{n+1}

that

|w_{n+1}| ≤ |w_{n}|(1 + c|y_{n} − x_{n}|)
+ c Min (c_{5} ρ^{1+1/2k}, |x_{n} − y_{n}| + |x̃_{n} − ỹ_{n}|),

starting with

|w_{n_{0}}| ≤ cc_{5} ρ^{1−1/2k}.

It is thus clear that |w_{n}| ≤ cc_{5} ρ^{1−1/2k} for all n ≥ n_{0} , which gives part (i) of Lemma §4.7. 5.3 We now turn to the second part of Lemma §4.7.

Let \underline{θ} ∈ Σ^{−}, and \underline{a} = a_{0} … a_{m} a word in Σ with a_{0} = θ_{0} and |I(\underline{a})| > c_{0}^{−1} ρ. We define

\begin{matrix}u_{−m} = ũ_{−m} = 0, & v_{−m} = ṽ_{−m} = 1 \\ u_{−j} = f_{a_{j}a_{j+1}}(u_{−j−1}), & v_{−j} = f_{a_{j}a_{j+1}}(v_{−j−1}) \\ ũ_{−j} = f̃_{a_{j}a_{j+1}}(ũ_{−j−1}), & ṽ_{−j} = f̃_{a_{j}a_{j+1}}(ṽ_{−j−1}), \end{matrix}

for 0 ≤ j < m. Then u_{0} , v_{0} (resp. ũ_{0} , ṽ_{0}) are the endpoints of I(\underline{a}) (resp. I\underline{ω}(\underline{a})). For n ≥ 0, we define (x_{n})_{n≥0} , (y_{n})_{n≥0} , (u_{n})_{n≥0} , (v_{n})_{n≥0} , (x̃_{n})_{n≥0} , (ỹ_{n})_{n≥0} , (ũ_{n})_{n≥0} , (ṽ_{n})_{n≥0} as in 5.2. One has

\frac{|I^{\underline{θ},\underline{ω}}(\underline{a})|}{|I\underline{θ}(\underline{a})|} = Lim_{n→+∞} \frac{|ṽ_{n} − ũ_{n}|}{|v_{n} − u_{n}|} \frac{|y_{n} − x_{n}|}{|ỹ_{n} − x̃_{n}|} ·

As above, we have

Max(|v_{n} − ṽ_{n}|, |u_{n} − ũ_{n}|) ≤ cc_{5} ρ^{1+1/2k}, ∀ n ≥ −m
Max(|x_{n} − x̃_{n}|, |y_{n} − ỹ_{n}|) ≤ cc_{5} ρ^{1+1/2k}, ∀ n ≥ 0.

Let n_{1} be the largest integer for which |v_{n} − u_{n}| ≥ c_{0}^{−3} ρ^{−1/k}; we have

| Log \frac{|ṽ_{n_{1}} − ũ_{n_{1}}|}{|v_{n_{1}} − u_{n_{1}}|}| ≤ cc_{5} ρ^{1−1/2k}

and, for n ≥ n_{1}

\frac{ṽ_{n+1} − ũ_{n+1}}{v_{n+1} − u_{n+1}} = \frac{ṽ_{n} − ũ_{n}}{v_{n} − u_{n}} λ_{n} ,

with

λ_{n} = \frac{∫_{0}^{1} Df_{n+1}(ũ_{n} + t(ṽ_{n} − ũ_{n}))dt}{∫_{0}^{1} Df_{n+1}(u_{n} + t(v_{n} − u_{n}))dt}

so that

|λ_{n} − 1| ≤ cc_{5} ρ^{1+1/2k}.

This means that for n_{2} = [ρ^{−1/k}] ≫ n_{1} , we will still have

| Log \frac{|ṽ_{n_{2}} − ũ_{n_{2}}|}{|v_{n_{2}} − u_{n_{2}}|}| ≤ cc_{5} ρ^{1−1/2k}.

We can show in the same way that

| Log \frac{|ỹ_{n_{2}} − x̃_{n_{2}}|}{|y_{n_{2}} − x_{n_{2}}|}| ≤ cc_{5} ρ^{1−1/2k}

For n ≥ 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}|} α_{n}

with |α_{n} − 1| ≤ c|y_{n} − x_{n}|; this allows to get

Lim_{n→+∞} \frac{|v_{n} − u_{n}|}{|y_{n} − x_{n}|} = \frac{|v_{n_{2}} − u_{n_{2}}|}{|y_{n_{2}} − x_{n_{2}}|} α,

with |α − 1| ≤ c|y_{n_{2}} − x_{n_{2}}| (because |y_{n} − x_{n}| is decreasing geometrically). Similarly

Lim_{n→+∞} \frac{|ṽ_{n} − ũ_{n}|}{|ỹ_{n} − x̃_{n}|} = \frac{|ṽ_{n_{2}} − ũ_{n_{2}}|}{|ỹ_{n_{2}} − x̃_{n_{2}}|} α̃,

with |α̃ − 1| ≤ c|ỹ_{n_{2}} − x̃_{n_{2}}|. As we have

Max(|y_{n_{2}} − x_{n_{2}}|, |ỹ_{n_{2}} − x̃_{n_{2}}|) ≪ ρ^{1−1/2k}.

we obtain the required estimate. 5.4 In the rest of this section, we will prove the Proposition §3.8.

We first recall the setting.

We are given u = (\underline{θ}, \underline{θ}′, s, t) ∈ L̃ × ℝ.

The set Ω_{1}(u) is the set of parameters \underline{ω} ∈ Ω, such that there exists ũ = (\underline{θ̃}, \underline{θ̃}′, s̃, t̃) ∈ L̃ × ℝ with

d(\underline{θ}, \underline{θ̃}) ≤ 2ρ^{5/2}, d(\underline{θ}′, \underline{θ̃}′) ≤ 2ρ^{5/2}, |s − s̃| < ρ, |t − t̃| < ρ

and t̃∈ L0 ; \underline{ω}(θ̃, θ̃′, s̃).

The set Ω_{0}(u) is the set of parameters \underline{ω} ∈ Ω for which there exists \underline{b} ∈ Σ(ρ), \underline{b}′ ∈ Σ′(ρ) (with b_{0} = θ_{0} , b′_{0} = θ′_{0}) such that the image

T \underline{ω} ; \underline{b}T′_{\underline{b}′}(u) = û = (\underline{θ̂}, \underline{θ̂}′, ŝ, t̂)

satisfies

(θ̂, θ̂′, ŝ′) ∈ L for |ŝ − ŝ′| < \frac{1}{2}ρ^{1/2}

t̂∈ L0 ; \underline{ω}(θ̂, θ̂′, ŝ).

We have to prove that, provided c_{5} is large enough

ℙ(Ω^{1}(u) − Ω^{0}(u)) ≤ exp(−c_{7} ρ^{− \frac{1}{2k} (d+d′−1)})

where ℙ is normalized Lebesgue measure on Ω. 5.5 Recall the decomposition of 4.5:

Ω = [−1, +1]^{Σ_{1}(\underline{θ})} × [−1, +1]^{Σ_{1}−Σ_{1}(\underline{θ})}

\underline{ω} = (\underline{ω}′, \underline{ω}′′)

which only depends on \underline{θ} through an endword in Σ(ρ^{1/2k}). We have set \underline{ω}^{∗} = (\underline{0}, \underline{ω}′′) in §4.5, and L0 ; \underline{ω}(\underline{θ}, \underline{θ}′, s) actually only depend on \underline{ω}^{∗} (or \underline{ω}′′), not on \underline{ω}′.

Therefore, the property \underline{ω} ∈ Ω^{1}(u) depends only on \underline{ω}′′ (one has Σ_{1}(\underline{θ̃}) = Σ_{1}(\underline{θ}) as d(\underline{θ}, \underline{θ̃}) ≤ 2ρ^{5/2}).

We will fix \underline{ω}′′ ∈ Ω′′ = [−1, +1]^{Σ_{1}−Σ_{1}(\underline{θ})}. Then either \underline{ω}^{∗} = (\underline{0}, \underline{ω}′′) ∉ Ω^{1}(u), and then ([−1, +1]^{Σ_{1}(\underline{θ})} × \{\underline{ω}′′\}) ∩ Ω^{1}(u) is empty, or \underline{ω}^{∗} ∈ Ω^{1}(u). In this last case, we will prove that

ℙ_{Ω′}(Ω′ − Ω^{′0}(u)) ≤ exp(−c_{7} ρ^{− \frac{1}{2k} (d+d′−1)}),

where

Ω′ = [−1, +1]^{Σ_{1}(\underline{θ})},
Ω^{′0}(u) = {ω′ ∈ Ω′, (ω′, ω′′) ∈ Ω^{0}(u)},

and ℙ_{Ω′} is Lebesgue measure normalized on Ω′. Proposition §3.8 will then follow by Fubini’s theorem.

5.6 Now on, \underline{ω}′′ is fixed, with \underline{ω}^{∗} ∈ Ω^{1}(\underline{u}). This means that there exists ũ = (θ̃, θ̃′, s̃, t̃) in L̃ × ℝ such that

d(\underline{θ}, \underline{θ̃}) ≤ 2ρ^{5/2}, d(\underline{θ}′, \underline{θ̃}′) ≤ 2ρ^{5/2}, |s − s̃| < ρ, |t − t̃| < ρ
and t̃∈ L_{\underline{ω}^{∗}}^{0}(θ̃, θ̃′, s̃).

By definition of L_{\underline{ω}^{∗}}^{0} , there exist pairs (\underline{b}^{1}, \underline{b}^{′1}), …, (\underline{b}^{N}, \underline{b}^{′N}) (with b_{0}^{i} = θ̃_{0} = θ_{0} , b_{0}^{′i} = θ̃′_{0} = θ′_{0}), with N = [c_{16}^{2} ρ^{− \frac{1}{2k} (d+d′−1)}], such that, writing

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{b} & ^{i} \end{matrix}T′_{\underline{b}^{′i}}(\underline{θ̃}, \underline{θ̃}′, s̃, t̃) =: ũ_{i}
=: (\underline{θ̃}^{i}, \underline{θ̃}^{′i}, s̃_{i}, t̃_{i})

we have (i) \underline{b}^{1}, …, \underline{b}^{N} are pairwise independent; (ii) \underline{θ̃}^{i} ∈ Σ_{nr}^{−} for 1 ≤ i ≤ N;

(iii) (\underline{θ̃}^{i}, \underline{θ̃}^{′i}, s̃′_{i}) ∈ L^{−} for 1 ≤ i ≤ N, |s̃_{i} − s̃′_{i}| ≤ \frac{2}{3}ρ^{1/2};

(iv) |t̃_{i}| ≤ 2(1 + R) for 1 ≤ i ≤ N. For 1 ≤ i ≤ N, consider

u_{i} = (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}, t_{i}) := T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{b} & ^{i} \end{matrix}T′_{\underline{b}^{′i}}(\underline{θ}, \underline{θ}′, s, t),

and more generally, for \underline{ω}′ ∈ Ω′, \underline{ω} = (\underline{ω}′, ω′′):

u_{i}(\underline{ω}′) = (\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}′), t_{i}(\underline{ω}′)) := T \underline{ω} ; \underline{b}^{i}T′_{\underline{b}^{′i}}(\underline{θ}, \underline{θ}′, s, t).

Lemma: We have, for 1 ≤ i ≤ N, \underline{θ}^{i} ∈ Σ_{nr}^{−} and, ∀ \underline{ω}′ ∈ Ω′, (\underline{θ}^{i}, \underline{θ}^{′i}, s′_{i}) ∈ L̃ if |s′_{i} − s_{i}(ω′)| <

\frac{1}{2}ρ^{1/2}.

Proof: From d(\underline{θ}, \underline{θ̃}) ≤ 2ρ^{5/2}, it follows that d(\underline{θ}^{i}, \underline{θ̃}^{i}) ≤ c ρ^{7/2}; but \underline{θ̃}^{i} ∈ Σ_{nr}^{−} and this set is determined by a condition on an endword in Σ(ρ^{3}). Therefore \underline{θ}^{i} ∈ Σ_{nr}^{−}. Moreover, L̃ has the same fiber over (\underline{θ̃}^{i}, \underline{θ̃}^{′i}). By proposition §1.3, k^{\underline{θ}^{i}} and k^{\underline{θ̃}^{i}} are at a C^{1} distance of order at most ρ^{5/2}, and similarly for k^{′\underline{θ}^{′i}} and k^{′\underline{θ̃}^{′i}}. Therefore

|s̃_{i} − s_{i}| ≤ c|s̃ − s| + c ρ^{5/2} ≤ c ρ.

Also, by Lemma 4.6, part (ii):

|s̃_{i}(\underline{ω}′) − s̃_{i}| ≤ cc_{5} c_{18} ρ^{1−1/2k} ≤ \frac{1}{10}ρ^{1/2}

if ρ is small enough. By property (iii) above, we therefore will have

(\underline{θ}^{i}, \underline{θ}^{′i}, s′_{i}) ∈ L̃ if |s′_{i} − s̃_{i}(\underline{ω}′)| < \frac{1}{2}ρ^{1/2}. □

5.7 Let \underline{ω}′ ∈ Ω′; in view of the lemma above, if there exists 1 ≤ i ≤ N with

t_{i}(\underline{ω}′) ∈ L0 ; \underline{ω}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}′))

then \underline{ω}′ ∈ Ω^{′0}(u). Let \underline{a} ∈ Σ_{2}^{−}, \underline{a} = (θ_{−r}, …, θ_{0}) be the endword of \underline{θ} in Σ_{2}^{−}(⊂ Σ(ρ^{1/2k})). For 1 ≤ i ≤ N, let

\underline{a}^{i} = (θ_{−r}, …, θ_{0} = b_{0}^{i}, b_{1}^{i}, …, b_{r_{i}}^{i})

be the element in Σ_{1}(\underline{θ}) obtained by concatenation of \underline{a} and the beginning of \underline{b}^{i}. Because \underline{b}^{1}, …, \underline{b}^{N} are pairwise independent, the elements \underline{a}^{1}, …, \underline{a}^{N} of Σ_{1}(\underline{θ}) are distinct.

Let Σ̃_{1}(\underline{θ}) = Σ_{1}(\underline{θ})−\{\underline{a}^{1}, …, \underline{a}^{N}\}; we will just denote by ω_{i} (1 ≤ i ≤ N) the coordinate of \underline{ω}′ corresponding to \underline{a}^{i}. We thus have

Ω′ = [−1, +1]^{N} × Ω̃′, Ω̃′ = [−1, +1]^{Σ̃_{1}(θ)}
\underline{ω}′ = ((ω_{1}, …, ω_{N}), \underline{ω̃}′).

We will prove that for any \underline{ω̃}′ ∈ Ω̃′, the set of (ω_{1}, …, ω_{N}) for which there exists 1 ≤ i ≤ N with t_{i}(\underline{ω}′) ∈ L0 ; \underline{ω}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{θ}′)) fills [−1, +1]^{N} except for a set of normalized Lebesgue measure < exp(−c_{7} ρ^{− \frac{1}{2k} (d+d′−1)}) (provided c_{5} is large enough). By Fubini’s theorem, this will imply what we want.

5.8 For \underline{ω̃}′ ∈ Ω̃′, set \underline{ω̂} = ((\underline{0}, \underline{ω̃}′), \underline{ω}′′).

For 1 ≤ i ≤ N, define

L^{i} := L−1 ; \underline{ω̂}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(ω̂)) (cf. §4.5).

Choose \underline{\overline{θ}} ∈ Σ^{−} with d(\underline{\overline{θ}}, \underline{θ}) < ρ^{5/2}, such that the endword \underline{a} ∈ Σ_{2}^{−} of \underline{θ} (and \underline{\overline{θ}}) does not appear elsewhere in \underline{\overline{θ}}. For \underline{ω}′ = (ω_{1}, …, ω_{N}, ω̃′), \underline{ω} = (\underline{ω}′, \underline{ω}′′), consider

T \underline{ω} ; \underline{b}^{i}T′_{\underline{b}^{′i}}(\underline{\overline{θ}}, \underline{θ}′, s, t) = (\underline{\overline{θ}}^{i}, \underline{θ}^{′i}, s̄_{i}(\underline{ω}′), t̄_{i}(\underline{ω}′)).

Lemma 1: t̄_{i}(\underline{ω}′) depends only (for fixed \underline{ω}′′) on ω_{i}, not on ω̃′ and ω_{j} , j ≠ i. Moreover, there exists c′_{7} > 0 such that

m{ω_{i}, t̄_{i}(ω_{i}) ∈ L^{i}} ≥ c′_{7} .

Lemma 2: Let \underline{ω}′ = (ω_{1}, …, ω_{N}, \underline{ω̃}′) ∈ Ω′ and 1 ≤ i ≤ N. If t̄_{i}(ω_{i}) ∈ L^{i}, then t_{i}(\underline{ω}′) ∈

L0 ; \underline{ω}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}′)), and thus \underline{ω}′ ∈ Ω′_{0}(u).

Clearly the two lemmas imply Proposition §3.8, if we recall that N = [c_{16}^{2} ρ^{−1/2k(d+d′−1)}].

5.9 Proof of Lemma 1.

Let 1 ≤ i ≤ N. We want to understand how the endpoints of K^{\underline{\overline{θ}},\underline{ω}}(\underline{b}^{i}) depend on \underline{ω}′. Consider first the sequence defined by

x_{0} = x_{0}(\underline{ω}′) = 0, x_{n}(\underline{ω}′) = f\underline{ω} ; \overline{θ}_{−n}\overline{θ}_{−n+1}(x_{n−1}(\underline{ω}′)) for n > 0.

Claim: for n ≥ 0, x_{n} does not depend on \underline{ω}′. By §3.6.3, it is sufficient to prove that for n > 0, x_{n}(\underline{ω}′) does not belong to any Î(\underline{ã}),

\underline{ã} ∈ Σ_{1}(\underline{θ}).

If |I(\overline{θ}_{−n}, …, \overline{θ}_{0})| ≤ ρ^{\frac{3}{5k}} this is true because elements of Σ_{1}(\underline{θ}) start with \underline{a}, and there is only one occurence of \underline{a} in \underline{\overline{θ}}: \underline{a} = (\overline{θ}_{−r}, …, \overline{θ}_{0}) ∈ Σ(ρ^{1/2k}).

If |I(\overline{θ}_{−n}, …, \overline{θ}_{0})| > ρ^{3/5k}, x_{n}(\underline{ω}′) cannot in fact belong to any Î(\underline{ã}), ã ∈ Σ_{1} , at least if ρ is small enough; indeed, the endpoints of K(θ_{0}) are preperiodic under g, so ã would end with a preperiodic word (independent of ρ) of approximate size ρ^{2/5k}, which precludes ã ∈ Σ_{1} . □

In the same way, one proves that the sequence

y_{0} = y_{0}(\underline{ω}′) = 1, y_{n}(\underline{ω}′) = f\underline{ω} ; \overline{θ}_{−n},\overline{θ}_{−n+1}(y_{n−1}, (\underline{ω}′)),

is independent of \underline{ω}′.

Consider now, with \underline{b}^{i} = (b_{0}^{i}, …, b_{m}^{i}), the sequence u_{−m} = u_{−m}(\underline{ω}′) = 0,

u_{−j}(\underline{ω}′) = f\underline{ω} ; b_{j}^{i},b_{j+1}^{i}(u_{−j−1}(\underline{ω}′)), 0 ≤ j < m

u_{n}(\underline{ω}′) = f\underline{ω} ; \overline{θ}_{1−n},\overline{θ}_{−n+1}(u_{n−1}(\underline{ω}′)), n > 0.

For the same reasons than above (using that \underline{a} does not appear twice in \underline{\overline{θ}}^{i}), we have that u_{n}(\underline{ω}′) is independent of \underline{ω}′ for −m ≤ n < r. Then we have, by §3.6.3

u_{r}(\underline{ω}′) = u_{r}(\underline{0}) + c_{5} ω_{i} ρ^{1+1/2k}.

For n > r, u_{n}(\underline{ω}′) again cannot belong to any Î(ã), ã ∈ Σ_{1}(\underline{θ}). Setting

\underline{θ^{=}} = σ^{−r}(\underline{\overline{θ}}) = (… \overline{θ}_{−r−1}, \overline{θ}_{−r})

we see that the endpoint of I^{\underline{\overline{θ}},\underline{ω}}(\underline{b}^{i}) determined by (u_{n}) satisfies

u(\underline{ω}′) = B ∘ k^{\underline{θ}^{=},ω̂}(u_{r}(0) + c_{5}ω_{i} ρ^{1+1/2k}),

where B is the affine map (with the appropriate orientation) sending I(\underline{a}) onto I(θ_{0}). Neither B nor k^{\underline{θ}^{=},ω̂} depend on \underline{ω}′.

The other endpoint of I^{\underline{\overline{θ}},\underline{ω}}(\underline{b}^{i}) satisfies a similar formula.

This means that t̄_{i}, as a function of \underline{ω}′, only depends on ω_{i}. Moreover, as k^{θ^{=},ω̂} is bounded in the C^{1}-topology and the derivative of B is of order ρ^{−1/2k}, we have

c^{−1}c_{5} < |\frac{∂t̄_{i}}{∂ω_{i}} | < cc_{5} .

On the other hand, we have |L_{i}| > c_{17} by proposition §4.5, and L_{i} ⊂ [−(1 + R), 1 + R]. So to end the proof of Lemma 1, we have just to control t̄_{i}(\underline{0}). Let

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{b} & ^{i} \end{matrix}T′_{\underline{b}^{′i}}(\underline{\overline{θ}}, \underline{θ}′, s, t) = (\underline{\overline{θ}}^{i}, \underline{θ}^{′i}, s_{i}^{∗}, t_{i}^{∗}).

We have just seen that in fact t_{i}^{∗} = t̄_{i}(0). On the other hand, we have

T\begin{matrix}\underline{ω} & ^{∗} \\ \underline{b} & ^{i} \end{matrix}T′_{\underline{b}^{′i}}(\underline{θ̃}, \underline{θ̃}′, s̃, t̃) = (\underline{θ̃}^{i}, \underline{θ̃}^{′i}, s̃_{i}, t̃_{i}),

with d(\underline{\overline{θ}}, \underline{θ̃}) ≤ 3ρ^{5/2}, d(\underline{θ}′, \underline{θ̂}′) ≤ 2ρ^{5/2}, |s − s̃| ≤ ρ, |t − t̃| < ρ and

|t̃_{i}| ≤ 2(1 + R).

Then |t̃_{i} − t_{i}^{∗}| ≤ cR, which implies

|t̄_{i}(0)| ≤ 2(1 + R) + cR;

this concludes the proof of Lemma 1.

5.10 Proof of Lemma 2.

Let \underline{ω}′ = (ω_{1}, …, ω_{N}, \underline{ω̃}′), \underline{ω} = (\underline{ω}′, \underline{ω}′′), \underline{ω̂} = ((0, ω̃′), \underline{ω}′′). Let also 1 ≤ i ≤ N. We assume that t̄_{i}(ω_{i}) ∈ L−1 ; \underline{ω̂}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(ω̂)) and we want to show that

t_{i}(\underline{ω}′) ∈ L0 ; \underline{ω}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}′)).

The hypothesis means that there exist pairs (\underline{c}^{1}, \underline{c}^{′1}), …, (\underline{c}^{N}, \underline{c}^{′N}) (with c_{0}^{j} = θ_{0}^{i}, c_{0}^{′j} = θ_{0}^{′i} in Σ(ρ) × Σ′(ρ) such that, if

T_{\underline{c}^{j}}^{\underline{ω̂}(i)}T\underline{c}^{′j}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(ω̂), t̄_{i}(ω_{i})) = (\underline{θ}^{(j)}, \underline{θ}^{′(j)}, ŝ_{(j)}, t̂_{(j)})

(where \underline{ω̂}(i) is obtained from ω̂ by setting the coordinates in Σ_{1}(\underline{θ}^{i}) equal to 0), one has (i) \underline{c}^{1}, …, \underline{c}^{N} are pairwise independent; (ii) \underline{θ}^{(j)} ∈ Σ_{nr}^{−} for 1 ≤ j ≤ N;

(iii)’ (\underline{θ}^{(j)}, \underline{θ}^{′(j)}, ŝ′_{(j)}) ∈ L̃ for |ŝ_{(j)} − ŝ′_{(j)}| < \frac{3}{4}ρ^{1/2};

(iv)’ |t̂_{(j)}| ≤ 1 + R for 1 ≤ j ≤ N.

Let

T_{\underline{c}^{j}}^{\underline{ω}(i)}T′_{\underline{c}^{′j}}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}′), t_{i}(\underline{ω}′)) = (\underline{θ}^{(j)}, \underline{θ}^{′(j)}, s_{(j)}, t_{(j)}),

where \underline{ω}(i) is obtained from \underline{ω} by setting the coordinates in Σ_{1}(\underline{θ}^{i}) equal to 0.

We will show tht for 1 ≤ j ≤ N:

(iii) (\underline{θ}^{(j)}, \underline{θ}^{′(j)}, s′_{(j)}) ∈ L̃ for |s′_{(j)} − s_{(j)}| < \frac{2}{3}ρ^{1/2},

(iv) |t_{(j)}| ≤ 2(1 + R), and this implies Lemma 2.

It is thus sufficient to show that

|s_{(j)} − ŝ_{(j)}| < \frac{1}{12}ρ^{1/2},

|t_{(j)} − t̂_{(j)}| < 1 + R.

The first inequality is easy: by Lemma §4.7, part (ii), we have

|s_{i}(\underline{ω}′) − s_{i}(ω̂)| ≤ c_{5}c_{18} ρ^{1−1/2k}

and applying a second time the same estimate from this lemma (to compare T_{\underline{c}^{j}}^{\underline{ω̂}(i)} and T_{\underline{c}^{j}}^{\underline{ω}(i)}) will give

|s_{(j)} − ŝ_{(j)}| < cc_{5}c_{18} ρ^{1−1/2k}

which is < \frac{1}{12}ρ^{1/2} for small enough ρ.

To prove the second inequality, we first compare t̄_{i}(ω_{i}) and t_{i}(\underline{ω}′). By definition, t̄_{i}(ω_{i}) is the t-coordinate of T \underline{ω} ; \underline{b}^{i}T\underline{b}^{′i}(\underline{\overline{θ}}, \underline{θ}′, s, t), while t_{i}(\underline{ω}′) is the t-coordinate of T \underline{ω} ; \underline{b}^{i}T\underline{b}^{′i}(\underline{θ}, \underline{θ}′, s, t). As d(\underline{θ}, \underline{\overline{θ}}) ≤ ρ^{5/2}, the C^{0}-distance between k\underline{θ}, k\underline{\overline{θ}} is at most of order ρ^{5/2}, and this imply

|t̄_{i}(ω_{i}) − t_{i}(\underline{ω}′)| ≤ ρ^{3/2}.

We next prove the

Claim: |s_{i}(\underline{ω̂}) − s_{i}(\underline{ω}′)| ≤ cc_{5} ρ^{1+1/2k} (a reforcement of Lemma §4.7 part (ii) in this case).

Proof: Let

T \underline{ω} ; \underline{b}^{i}T′_{\underline{b}^{′i}}(\underline{\overline{θ}}, \underline{θ}′, s, t) = (\underline{\overline{θ}}^{i}, \underline{θ}^{′i}, s̄_{i}(\underline{ω}′), t̄_{i}(\underline{ω}′)).

As d(\underline{θ}, \underline{\overline{θ}}) ≤ ρ^{5/2}, k^{\underline{θ},\underline{ω}} and k^{\underline{\overline{θ}},\underline{ω}} are at a C^{1}-distance of order no more than ρ^{5/2}. This implies that

|s_{i}(\underline{ω̂}) − s̄_{i}(\underline{ω̂})| ≤ c ρ^{5/2}
|s_{i}(\underline{ω}′) − s̄_{i}(\underline{ω}′)| ≤ c ρ^{5/2}.

To compare \overline{s}_{i}(\underline{ω̂}), s̄_{i}(\underline{ω}′), we go back to the proof of Lemma 1. The endpoints of I^{\underline{\overline{θ}},\underline{ω}}(\underline{b}^{i}) depend, for fixed \underline{ω}′′, only on ω_{i}:

u(\underline{ω}′) = B ∘ k^{\underline{θ^{=}},ω̂}(u_{r}(0) + c_{5}ω_{i} ρ^{1+1/2k}),

v(\underline{ω}′) = B ∘ k^{\underline{θ^{=}},ω̂}(v_{r}(0) + c_{5}ω_{i} ρ^{1+1/2k}).

As k^{\underline{θ^{=}},ω̂} is C^{2}-bounded, we obtain

|s̄_{i}(\underline{ω̂}) − s̄_{i}(\underline{ω}′)| ≤ cc_{5} ρ^{1+1/2k},

proving the claim. □

Finally, let

T_{\underline{c}^{j}}^{\underline{ω̂}(i)} T′_{\underline{c}^{′j}}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω}′), t_{i}(\underline{ω}′)) = (\underline{θ}^{(j)}, \underline{θ}^{′(j)}, s̃_{(j)}, t̃_{(j)}).

Lemma 3: We have |t_{(j)} − t̃_{(j)}| < cc_{5} ρ^{1/2k}. The lemma is proved in the next section. Using it, we finish the proof of Lemma 2. Let t^{≈}_{(j)} be the t-coordinate of

T_{\underline{c}^{j}}^{\underline{ω̂}(i)} T′_{\underline{c}^{′j}}(\underline{θ}^{i}, \underline{θ}^{′i}, s_{i}(\underline{ω̂}), t_{i}(\underline{ω}′)).

Because |t̄_{i}(ω_{i}) − t_{i}(\underline{ω}′)| ≤ c ρ^{3/2}, we have

|t̂_{(j)} − t^{≈}_{(j)}| ≤ c ρ^{1/2}.

Because |s_{i}(\underline{ω}′) − s_{i}(\underline{ω̂})| ≤ cc_{5} ρ^{1+1/2k}, we have

| t^{≈}_{(j)} − t̃_{(j)}| ≤ cc_{5} ρ^{\frac{1}{2k}} .

By Lemma 3, the same estimate holds for |t̃_{(j)} − t_{(j)}|. We conclude that for ρ small enough, we have indeed

|t_{(j)} − t̂_{(j)}| < 1 + R,

as was required. □

5.11 Proof of Lemma 3.

We will compare, for s̄ ∈ J_{R} and t̄∈ ℝ, the t-coordinates of

T_{\underline{c}^{j}}^{\underline{ω}(i)} T′_{\underline{c}^{′j}}(\underline{θ}^{i}, \underline{θ}^{′i}, s̄, t̄)

and

T_{\underline{c}^{j}}^{\underline{ω̂}(i)} T′_{\underline{c}^{′j}}(\underline{θ}^{i}, \underline{θ}^{′i}, s̄, t̄).

The two elements \underline{ω}(i), \underline{ω̂}(i) have different coordinates only at \underline{a}^{1}, …, \underline{a}^{N} where those of \underline{ω}(i) are arbitrary and those of \underline{ω̂}(i) are 0. We observe also that, because \underline{θ}^{i} ∈ Σ_{nr}^{−}, we have

Σ_{1}(\underline{θ}) ∩ Σ_{1}(\underline{θ}^{i}) = ∅,

and that the coordinates of \underline{ω}(i) (or \underline{ω̂}(i)) in Σ_{1}(\underline{θ}^{i}) are 0.

Thus, we have to understand how the t-coordinate of T_{\underline{c}^{j}}^{\underline{ω}(i)} T′_{\underline{c}^{′j}}(\underline{θ}^{i}, \underline{θ}^{′i}, s̄, t̄) depends on (ω_{1}, …, ω_{N}), which amounts to understand how the endpoints of I^{\underline{θ}^{i},\underline{ω}(i)}(\underline{c}^{j}) depend on (ω_{1}, …, ω_{N}).

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

In the iteration process of §5.9, again the only time where things depend on (ω_{1}, …, ω_{N}) is when we apply f_{\overline{θ}_{−r} \overline{θ}_{−r+1}}^{\underline{ω}(i)} (we use here that \underline{θ}^{(j)} ∈ Σ_{nr}^{−}). We will see as in §5.9 that the

endpoints w^{±} of I^{\underline{\overline{θ}}^{i},\underline{ω}(i)}(\underline{c}^{j}) do not depend on ω_{k}, k ≠ i, and the dependence on ω_{i} is as follows

w^{±}(ω_{i}) = \frac{k̄(w_{r}^{±} + c_{5}ω_{i}ρ^{1+1/2k}) − k̄(u_{r} + c_{5}ω_{i}ρ^{1+1/2k})}{k̄(v_{r} + c_{5}ω_{i}ρ^{1+1/2k}) − k̄(u_{r} + c_{5}ω_{i}ρ^{1+1/2k})} ,

where k̄ = k^{\underline{θ^{=}},\underline{ω̂}(i)} does not depend on ω_{1}, …, ω_{N} . Writing

w^{+}(ω_{i}) = \frac{w_{r}^{+} − u_{r}}{v_{r} − u_{r}} \frac{∫_{0}^{1} Dk̄(u_{r} + t(w_{r}^{+} − u_{r}) + c_{5}ω_{i}ρ^{1+1/2k})dt}{∫_{0}^{1} Dk̄(u_{r} + t(v_{r} − u_{r}) + c_{5}ω_{i}ρ^{1+1/2k})dt} ,

we see that

|\frac{∂w^{+}(ω_{i})}{∂ω_{i}} | ≤ cc_{5} ρ^{1+1/2k},

as k̄ is C^{2}-bounded. We thus have the control on the endpoints of I^{\underline{\overline{θ}}^{i},\underline{ω}(i)}(\underline{c}^{j}) which guarantees the estimate of the lemma. □

5.12 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 ≤ m ≤ M, the same proof than above (with the \overline{L}_{ω} instead of L_{ω}) shows that

ℙ(\overline{Ω}^{1}(u) − Ω_{m}^{0}(u)) ≤ exp ( − c^{=}_{7} ρ^{− \frac{1}{2k} (d+d′−1)})

where in the definition of Ω_{m}^{0}(u) we require (\underline{b}, \underline{b}′) to be one the \underline{b}_{m}^{ℓ}, \underline{b}_{m′}^{ℓ′} . Thus we obtain

ℙ(\overline{Ω}^{1}(u) − \overline{Ω}^{0}(u)) ≤ M exp ( − c^{=}_{7} ρ^{− \frac{1}{2k} (d+d′−1)}) ≤ exp ( − c̄_{7} ρ^{− \frac{1}{2} (d+d′−1)}).

§6. Proof of the Scale recurrence Lemma.

6.1 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 Λ with a map Λ ↦ A × A

λ ↦ (α(λ), ω(λ)).

Define

Λ_{i}^{j} = {λ, α(λ) = i, ω(λ) = j},
Λ_{i} = {λ, α(λ) = i},
Λ^{j} = {λ, ω(λ) = j},
Λ^{(2)} = {(λ, λ′) ∈ Λ^{2}, α(λ′) = ω(λ)}.

Set also

N_{i}^{j} = # Λ_{i}^{j}, N_{i} = # Λ_{i} , p_{i}^{j} = N_{i}^{−1} N_{i}^{j} .

We will assume that there exists p^{∗} > 0 such that p_{i}^{j} ≥ p^{∗} for all i, j.

The stochastic matrix (p_{i}^{j}) has a left eigenvector (p^{i})_{i∈A} satisfying

∑_{i} p^{i}p_{i}^{j} = p^{j} ≥ p^{∗}.
∑_{i} p^{i} = 1.

If we set

p_{λ}^{λ′} = \{ \begin{matrix}0 if ω(λ) ≠ α(λ′) \\ N_{i}^{−1} if ω(λ) = α(λ′) = i ∈ A, \end{matrix}

p^{λ} = N_{i}^{−1}p^{i}, λ ∈ Λ_{i}

then (p_{λ}^{λ′})_{(λ,λ′)∈Λ^{2}} is again a stochastic matrix with left eigenvector (p^{λ}). Indeed we have (with λ′ ∈ A, α(λ′) = j)

∑_{λ} p^{λ}p_{λ}^{λ′} = ∑_{i} ∑_{λ∈Λ_{i}^{j}} N_{i}^{−1} p^{i} N_{j}^{−1}

= ∑_{i} p_{i}^{j} p^{i} N_{j}^{−1} = p^{j} N_{j}^{−1} = p^{λ′}.

For z = (z_{λ})_{λ∈Λ} ∈ ℂ^{Λ}, define

||z||^{2} = ∑_{Λ} p^{λ}|z_{λ}|^{2}

= ∑_{i} p^{i} \frac{1}{N_{i}} ( ∑_{Λ_{i}} |z_{λ}|^{2}).

Observe that, if (θ_{λ}^{λ′}) is a family of complex numbers of modulus ≤ 1 and we define

w_{λ} = ∑_{λ′} p_{λ}^{λ′} θ_{λ}^{λ′} z_{λ′}

we have

|w_{λ}|^{2} ≤ ∑_{λ′} p_{λ}^{λ′} ∑_{λ′} p_{λ}^{λ′}|z_{λ′}|^{2} = ∑_{λ′} p_{λ}^{λ′}|z_{λ′}|^{2}

hence

||w||^{2} ≤ ||z||^{2}.

6.2 In the setting above, we consider a family (E_{λ})_{λ∈Λ} of bounded measurable subsets of ℝ, and a family a_{λ}^{λ′} of real numbers. Let also ρ > 0 be a small positive number. For λ ∈ Λ, x ∈ ℝ, ω(λ) = i, let

n_{λ}(x) = N_{i}^{−1} # {λ′ ∈ Λ_{i} , [x − ρ, x + ρ] ⊂ E_{λ′} − a_{λ}^{λ′}}.

We will assume that the following holds: there are Δ_{0} > 0, 0 < κ_{0} < 1 such that for any ξ ∈ [1, Δ_{0}ρ^{−1}], the operator

w_{λ} = ∑_{λ′} p_{λ}^{λ′} e^{iξa_{λ}^{λ′}} z_{λ′}

acting on (ℂ^{Λ}, || ||) has norm ≤ κ_{0} (we know already that the norm is ≤ 1).

Proposition: Under this hypothesis, there exists 0 < κ_{1} < 1, ε > 0, 0 < τ < 1, depending only on Δ_{0}, κ_{0} such that, if |E_{λ}| ≤ ε for all λ ∈ Λ, then

∑ p^{λ}|E_{λ}^{∗}| ≤ κ_{1} ∑ p^{λ}|E_{λ}|

where

E_{λ}^{∗} = {n_{λ}(x) > τ}.

6.3 Proof of proposition.

For λ ∈ Λ, ω(λ) = i, let:

X_{λ} = 1_{E_{λ}} ,

Y_{λ} = \frac{1}{N_{i}} ∑_{λ′∈Λ_{i}} X_{λ′}(x + a_{λ}^{λ′}),

Z_{λ}(x) = \frac{1}{2ρ} ∫_{−ρ}^{ρ} Y_{λ}(x + t) dt.

We have Z_{λ} ≥ n_{λ} hence

∑_{λ} p^{λ}|Z_{λ}|_{L^{2}}^{2} ≥ τ^{2} ∑_{λ} p^{λ}|E_{λ}^{∗}|.

We will show, under the hypotheses of the proposition that

∑_{λ} p^{λ}|Z_{λ}|_{L^{2}}^{2} ≤ κ_{2} ∑ p^{λ}|X_{λ}|_{L^{2}}^{2} = κ_{2} ∑_{λ} p^{λ}|E_{λ}|,

for some κ_{2} < 1 depending only on κ_{0}, Δ_{0} and this will imply the proposition (taking κ_{1} = τ = κ_{2}^{1/3}). To prove this inequality, we will consider the Fourier transforms of X_{λ}, Z_{λ}, Y_{λ}. We normalize the Fourier transform as

X̂_{λ}(ξ) = ∫_{−∞}^{+∞} X_{λ}(x)e^{−iξx} dx.

We then have

Ŷ_{λ}(ξ) = \frac{1}{N_{i}} ∑_{λ′∈Λ_{i}} X̂_{λ′}(ξ) e^{ia_{λ}^{λ′} ξ}

= ∑_{λ′} p_{λ}^{λ′} e^{ia_{λ}^{λ′} ξ} X̂_{λ′}(ξ),

and

Ẑ_{λ}(ξ) = \frac{sin ρξ}{ρξ} Ŷ_{λ}(ξ).

We estimate ∑ p^{λ}|Ẑ_{λ}(ξ)|^{2} in various ways, depending on ξ:

a) if |ξ| ≤ 1, we use

|Ẑ_{λ}(ξ)| ≤ |Ŷ_{λ}(ξ)| ≤ ∫_{−∞}^{+∞} Y_{λ}(x) dx

= ∑_{λ′} p_{λ}^{λ′}|E_{λ′}|

to get

∑_{λ} p^{λ}|Ẑ_{λ}(ξ)|^{2} ≤ ∑_{λ} p^{λ}( ∑_{λ′} p_{λ}^{λ′}|E_{λ′}|^{2})

≤ ∑_{λ′} p^{λ′}|E_{λ′}|^{2} ;

b) if 1 ≤ |ξ| ≤ Δ_{0}ρ^{−1}, we use again

|Ẑ_{λ}(ξ)| ≤ |Ŷ_{λ}(ξ)|

and by the hypothesis of the proposition ∑ p^{λ}|Ŷ_{λ}(ξ)|^{2} ≤ κ_{0}^{2} ∑ p^{λ}|X̂_{λ}(ξ)|^{2};

c) if |ξ| ≥ Δ_{0}ρ^{−1}, from 6.1, we have ∑ p^{λ}|Ŷ_{λ}(ξ)|^{2} ≤ ∑ p^{λ}|X̂_{λ}(ξ)|^{2},

but now

|\frac{sin ρξ}{ρξ} | ≤ κ_{3} = Max_{u≥Δ_{0}} |\frac{sin u}{u}| < 1,

hence ∑ p^{λ}|Ẑ_{λ}(ξ)|^{2} ≤ κ_{3}^{2} ∑ p^{λ}|X̂_{λ}(ξ)|^{2}. Putting these estimates together gives

∫_{−∞}^{+∞} ∑_{λ} p^{λ}|Ẑ_{λ}(ξ)|^{2} dξ ≤ [Max(κ_{0}, κ_{3})]^{2} ∫_{−∞}^{+∞} ∑_{λ} p^{λ}|X̂_{λ}(ξ)|^{2} dξ
+ 2 ∑ p^{λ}|E_{λ}|^{2}.

Now we have

||Z_{λ}||_{L^{2}}^{2} = \frac{1}{2π} ||Ẑ_{λ}||_{L^{2}}^{2}

||X_{λ}||_{L^{2}}^{2} = |E_{λ}| = \frac{1}{2π} ||X̂_{λ}||_{L^{2}}^{2} ,

and therefore

∑_{λ} p^{λ}|Z_{λ}|_{L^{2}}^{2} ≤ κ_{2} ∑ p^{λ}|E_{λ}|,

with κ_{2} = [Max(κ_{0}, κ_{3})]^{2} + π^{−1} ε. If ε is small enough, we have κ_{2} < 1 which concludes the proof of the proposition. □

Remarks: The main point of the proposition is that τ, ε, κ_{1} depend neither on the E_{λ} nor on the a_{λ}^{λ′} or the combinatorics.

6.4 In the same setting that 6.2, let Δ_{1} > 0 and

n̂_{λ}(x) = N_{i}^{−1} #{λ′ ∈ Λ_{i}, [x − Δ_{1}ρ, x + Δ_{1}ρ] ∩ (E_{λ′} − a_{λ}^{λ′}) ≠ ∅}.

We denote by V_{δ}(E) the δ-neighbourhood of a subset E ⊂ ℝ (i.e. the set of points at distance < δ from E).

Corollary: There exists 0 < κ_{4} < 1, ε_{1} > 0, 0 < τ < 1, depending only on Δ_{0} , κ_{0}, and there exists Δ > 0, depending only on Δ_{0} , Δ_{1} , κ_{0} such that, if |V_{Δρ}(E_{λ})| ≤ ε_{1} for all λ ∈ Λ, then

∑_{λ} p^{λ}|V_{Δρ}(Ê_{λ})| ≤ κ_{4} ∑_{λ} p^{λ}|V_{Δρ}(E_{λ})|,

where Ê_{λ} = {n̂_{λ} > τ}.

Proof: We first observe that, for any bounded subset E,

|V_{(Δ+Δ_{1}+1)ρ}(E)| ≤ (1 + Δ^{−1}(Δ_{1} + 1))|V_{Δρ}(E)|.

On the other hand, if [x−Δ_{1}ρ, x+Δ_{1}ρ] intersects E_{λ′} −a_{λ}^{λ′} and |y−x| < Δρ, then [y−ρ, y+ρ] is contained in V_{(Δ+Δ_{1}+1)ρ}(E_{λ′}) − a_{λ}^{λ′} .

We thus take Δ large enough to have

κ_{4} = (1 + Δ^{−1}(Δ_{1} + 1))κ_{1} < 1, ε_{1} = ε(1 + Δ^{−1}(Δ_{1} + 1))^{−1},

and apply the proposition to the V_{(Δ+Δ_{1}+1)ρ}(E_{λ}). □

6.5 After these general statements, we come back to the setting of the Scale recurrence Lemma.

Recall that Σ(ρ) is the set of words \underline{a} in Σ such that

c_{0}^{−1} ρ ≤ |I(\underline{a})| ≤ c_{0} ρ

and that we write

J_{R} = [−R, −R^{−1}] ∪ [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 ∈ A, a′ ∈ A′). Equivalently, all periodic points of g 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 §6.1, we set A = 𝔸 × 𝔸′ × {−1, +1},

Λ = Σ(ρ) × Σ′(ρ) × {−1, +1}.

The map (α, ω): Λ → A × A is defined as follows: for \underline{a} ∈ Σ(ρ), \underline{a}′ ∈ Σ′(ρ), u ∈ \{−1, +1\} α(\underline{a}, \underline{a}′, u) = (a_{0}, a′_{0}, uw)

ω(\underline{a}, \underline{a}′, u) = (a_{n}, a′_{n′} , u),

where a_{n}, a′_{n′} are the last letters of \underline{a}, \underline{a}′ and w = +1 (resp. −1) if f\underline{a}, f′_{\underline{a}′} have the same (resp. opposite) orientation.

It follows from the hypothesis of case B), and the hypothesis that Σ, Σ′ are topologically mixing, that if c_{0} has been chosen large enough we have, for all i, j ∈ A

c^{−1} ρ^{−(d+d′)} ≤ # Λ_{i}^{j} = N_{i}^{j} ≤ cρ^{−(d+d′)}.

We are given a family E(\underline{a}, \underline{a}′) of subsets of J_{R}, for \underline{a} ∈ Σ(ρ), \underline{a}′ ∈ Σ′(ρ). Define, for (\underline{a}, \underline{a}′, u) ∈ Λ

F (\underline{a}, \underline{a}′, u) = \{x, ue^{x} ∈ E(\underline{a}, \underline{a}′)\},

which is a subset of [−r, r], r = Log R. To say that all J_{R} − E(\underline{a}, \underline{a}′) have small measure amounts to say that all [−r, r] − F (λ), λ ∈ Λ, have small measure:

m([−r, r] − F (λ)) ≤ c̄_{1} .

Finally, we define the family of translations (a_{λ}^{λ′})_{(λ,λ′)∈Λ^{(2)}} of §6.2. To do this, we choose, for each \underline{a} ∈ Σ(ρ) (resp. \underline{a}′ ∈ Σ′(ρ)) an element \underline{θ} ∈ Σ^{−} (resp. \underline{θ}′ ∈ Σ^{′−}) ending with \underline{a} (resp. \underline{a}′). Then, if λ = (\underline{a}, \underline{a}′, u), λ′ = (\underline{b}, \underline{b}′, v) satisfy ω(λ) = α(λ′), we set

a_{λ}^{λ′} = Log |I^{\underline{θ}′}(\underline{b}′)| − Log |I\underline{θ}(\underline{b})|.

We will prove later the following

Lemma: The hypothesis of the Scale recurrence Lemma implies that the hypothesis of Proposition 6.2 holds, namely there exist Δ_{0} > 0, 0 < κ_{0} < 1 such that for ξ ∈ [1, Δ_{0}ρ^{−1}] the operator defined by

w_{λ} = ∑ p_{λ}^{λ′} e^{ia_{λ}^{λ′} ξ} z_{λ′}

has norm ≤ κ_{0} .

6.6 Let Δ_{1} > 0, and let then Δ be given by the corollary in 6.4. For λ ∈ Λ, we set

E_{0}(λ) = [−r, r] − V_{Δρ}(F (λ)).

Starting from E_{0} , we define for n ≥ 0 sets E_{n}(λ) in the following way; if E_{n}(λ) has already been defined, we define, with c_{3} > 0 fixed sufficiently small, ˆ Ê_{n}(λ) as the set of x ∈ [−r, r] such that

#{λ′, ω(λ) = α(λ′), [x + a_{λ}^{λ′} − Δ_{1}ρ, x + a_{λ}^{λ′} + Δ_{1}ρ] ⊂ [−r, r] − E_{n}(λ′)}

is less than c_{3} ρ^{−(d+d′)}. We then set

E_{n+1}(λ) = E_{0}(λ) ∪ ˆ Ê_{n}(λ).

We will see that, provided c̄_{1} , c_{3} are sufficiently small, we have

Lemma: ∑_{λ} p^{λ}|V_{Δρ}( ˆÊ_{n}(λ))| ≤ κ_{5} ∑_{λ} p^{λ}|V_{Δρ}(E_{n}(λ))|, with 0 < κ_{5} < 1.

Before proving the lemma we finish the proof of the Scale recurrence Lemma. It is clear that for each λ ∈ Λ, the sequence (E_{n}(λ))_{n≥0} is increasing. On the other hand, by the lemma, we have

∑ p^{λ}|V_{Δρ}(E_{n+1}(λ))| ≤ κ_{5} ∑ p^{λ}|V_{Δρ}(E_{n}(λ))| + Max_{λ}|V_{Δρ}(E_{0}(λ))|.

Defining E_{∞}(λ) = ⋃_{n≥0} E_{n}(λ), we thus obtain

∑ p^{λ}|V_{Δρ}(E_{∞}(λ))| ≤ \frac{1}{1 − κ_{5}} Max_{λ}|V_{Δρ}(E_{0}(λ))|.

We observe that V_{Δρ}(E_{0}(λ)) is contained in [−r − Δρ, r + Δρ] − F (λ), hence its measure is less than 2Δρ + c̄_{1} . Consider

F_{∞}(λ) = [−r, +r] − E_{∞}(λ).

Let x ∈ F_{∞}(λ); by definition of ˆ Ê_{n}(λ) ⊂ E_{n+1}(λ), we have, for every n ≥ 0

#{λ′, ω(λ) = α(λ′), [x + a_{λ}^{λ′} − Δ_{1}ρ, x + a_{λ}^{λ′} + Δ_{1}ρ] ⊂ [−r, r] − E_{n}(λ′)} ≥ c_{3}ρ^{−(d+d′)}

and therefore

#{λ′, ω(λ) = α(λ′), [x + a_{λ}^{λ′} − Δ_{1}ρ, x + a_{λ}^{λ′} + Δ_{1}ρ] ⊂ F_{∞}(λ′)} ≥ c_{3}ρ^{−(d+d′)}.

Now we come back to J_{R}, setting, for \underline{a} ∈ Σ(ρ), \underline{a}′ ∈ Σ′(ρ)

E^{∗}(\underline{a}, \underline{a}′) = \{e^{x}, x ∈ F_{∞}(\underline{a}, \underline{a}′, +1)\} ∪ \{−e^{x}, x ∈ F_{∞}(\underline{a}, \underline{a}′, −1)\}.

Observe that F_{∞}(λ) is contained in V_{Δρ}(F (λ)); this implies conclusion (i) of the Scale recurrence Lemma.

For conclusion (ii), as \underline{θ} → k\underline{θ} is Lipschitz, there exists C > 0 such that if λ = (\underline{a}, \underline{a}′, u), λ′ = (\underline{b}, \underline{b}′, v) satisfy ω(λ) = α(λ′) and \underline{θ}, \underline{θ′} are any elements in Σ^{−}, Σ^{′−} ending with \underline{a}, \underline{a}′ respectively, then

|Log |I^{\underline{θ}′}(\underline{b}′)| − Log |I\underline{θ}(\underline{b})| − a_{λ}^{λ′} | ≤ Cρ.

Taking Δ_{1} > 2C, for x ∈ F_{∞}(λ), we will have

x + Log |I^{\underline{θ}′}(\underline{b}′)| − Log |I\underline{θ}(\underline{b})| + z ∈ F_{∞}(λ′)

for |z| ≤ \frac{Δ_{1}}{2} ρ and at least c_{3}ρ^{−(d+d′)} elements λ′ (with α(λ′) = ω(λ)), for all \underline{θ}, \underline{θ}′ ending with \underline{a}, \underline{a}′ respectively. This immediately implies conclusion (ii) of the Scale recurrence Lemma.

6.7 We now prove the lemma at the end of §6.5.

Let η_{0} > 0 be extremely small, and ξ ∈ [1, Δ_{0} ρ^{−1}]. If the norm of the operator in the lemma is ≥ (1 − η_{0})^{1/2}, we can find z ∈ ℂ^{∧} with

||z||^{2} = Σ p^{λ}|z_{λ}|^{2} = 1,
||w||^{2} ≥ 1 − η_{0} ,

where

w_{λ} = ∑_{λ′} p_{λ}^{λ′} exp(iξ a_{λ}^{λ′})z_{λ′} .

Now, we have

|w_{λ}|^{2} ≤ ∑_{λ′} p_{λ}^{λ′}|z_{λ′}|^{2},

and all p^{λ} have order ρ^{d+d′}. Therefore there exists η_{1} > 0 very small, and Λ̃ ⊂ Λ, with

# (Λ − Λ̃) ≤ η_{1} ρ^{−(d+d′)},

|w_{λ}|^{2} ≥ (1 − η_{1}) ∑_{λ′} p_{λ}^{λ′}|z_{λ′}|^{2}, for all λ ∈ Λ̃.

Let λ ∈ Λ̃, ω(λ) = i; for λ′ ∈ Λ_{i}, set

Z_{λ}^{λ′} = exp(i a_{λ}^{λ′} ξ)z_{λ′} .

We have

−|w_{λ}|^{2} + ∑_{Λ_{i}} p_{λ}^{λ′}|z_{λ′}|^{2} = \frac{1}{2} ∑_{Λ_{i}} ∑_{Λ_{i}} p_{λ}^{λ′_{0}} p_{λ}^{λ′_{1}}|Z_{λ}^{λ′_{0}} − Z_{λ}^{λ′_{1}}|^{2},

hence

∑_{Λ_{i}} ∑_{Λ_{i}} |Z_{λ}^{λ′_{0}} − Z_{λ}^{λ′_{1}}|^{2} ≤ η_{2} ρ^{−2(d+d′)}

with very small η_{2} > 0, as p_{λ}^{λ′} has order ρ^{d+d′}. On the other hand ∑_{Λ_{i}} |Z_{λ}^{λ′}|^{2} has order ρ^{−(d+d′)}. Thus, if we write

Z_{λ} = N_{i}^{−1} ∑_{Λ_{i}} Z_{λ}^{λ′} ,

z_{λ′} = exp(−i a_{λ}^{λ′} ξ)Z_{λ} + Z̃_{λ}^{λ′} ,

we will have |Z_{λ}| of order 1 and

∑_{Λ_{i}} |Z̃_{λ}^{λ′}|^{2} ≤ η_{3} ρ^{−(d+d′)},

with very small η_{3} > 0.

If we take now λ_{0}, λ_{1} ∈ Λ̃ ∩ Λ^{i}, and compare the two expressions for z_{λ′} that we get, we obtain

∑_{Λ_{i}} | exp(i a_{λ_{0}}^{λ′} ξ) − exp(i a_{λ_{1}}^{λ′} ξ)|^{2} ≤ η_{4} ρ^{−(d+d′)},

meaning that

|sin [\frac{1}{2}(a_{λ_{0}}^{λ′} − a_{λ_{1}}^{λ′})ξ]| < η_{5}

holds for all λ′ ∈ Λ_{i} but at most η_{5} ρ^{−(d+d′)} elements. The constants η_{4} , η_{5} here are still small.

To derive a contradiction, we will choose λ_{0} , λ_{1} of the form

λ_{0} = (\underline{a}^{0}, \underline{a}′, u), λ_{1} = (\underline{a}^{1}, \underline{a}′, u)

which gives, with λ′ = (\underline{b}, \underline{b}′, v):

a_{λ_{0}}^{λ′} = Log |I^{\underline{θ̃}^{1}}(\underline{b})| − Log |I^{\underline{θ̃}^{0}}(\underline{b})|
= Log D[k^{\underline{θ̃}^{1}} ∘ (k^{\underline{θ̃}^{0}})^{−1}](y)

where \underline{θ̃} ends with \underline{a}^{i} and y ∈ I^{\underline{θ̃}^{0}}(\underline{b}).

It is now time to remember the hypothesis in the Scale recurrence Lemma: there exists \underline{θ}^{0}, \underline{θ}^{1} ∈ Σ^{−} (with the same last letter) and x ∈ K^{\underline{θ}^{0}}(θ_{0}^{0}), such that

D Log D[k^{\underline{θ}^{1}} ∘ (k^{\underline{θ}^{0}})^{−1}](x) = γ ≠ 0.

In a neighbourhood of \underline{θ}^{0} , \underline{θ}^{1} , x, this will still belongs to [\frac{1}{2}γ, \frac{3}{2}γ]. As #(Λ−Λ̃) ≤ η_{1} ρ^{−(d+d′)},

we can find \underline{a}^{0}, \underline{a}^{1}, \underline{a}′, u such that λ_{i} = (\underline{a}^{i}, \underline{a}′, u) ∈ Λ̃ and \underline{θ̃}^{i} is close to \underline{θ}^{i}. The number of λ′ = (\underline{b}, \underline{b}′, v) such that I^{\underline{θ̃}^{0}}(\underline{b}) is close to x is still of order ρ^{−(d+d′)}, and for two such λ′_{0} , λ′_{1} we will have

|(a_{λ_{0}}^{λ′_{0}} − a_{λ_{1}}^{λ′_{0}}) − (a_{λ_{0}}^{λ′_{1}} − a_{λ_{1}}^{λ′_{1}}) − γ(y_{0} − y_{1})| ≤ \frac{1}{2}|γ| |y_{0} − y_{1}|,

where y_{i} ∈ I^{\underline{θ̃}^{0}}(\underline{b}^{i}), \underline{b}^{i} being the first component of λ′_{1} .

On the other hand, for most λ′ we should have

|a_{λ_{0}}^{λ′} − a_{λ_{1}}^{λ′} − 2kπξ^{−1}| ≤ η_{6} ξ^{−1}

for some integer k and small η_{6} > 0. Combining the two inequalities, we should have for most choices of \underline{b}^{0} , \underline{b}^{1} (with I^{\underline{θ̃}^{0}}(\underline{b}) close to x) either |y_{0}−y_{1}| ≥ πξ^{−1}|γ|^{−1} or |y_{0}−y_{1}| ≤ 4η_{6} ξ^{−1}|γ|^{−1}. We have here 1 ≤ ξ ≤ Δ_{0}ρ^{−1}, hence πξ^{−1}|γ|^{−1} ≥ π|γ|^{−1} Δ_{0}^{−1}ρ. Taking Δ_{0} ≪ |γ| and η_{6} small enough contradicts the bounded geometry of regular Cantor sets.

6.8 We finally will prove the estimate

∑ p^{λ}|V_{Δρ}( ˆÊ_{n}(λ))| ≤ κ_{5} ∑ p^{λ}|V_{Δρ}(E_{n}(λ))|

of 6.6.

Recall that ˆ Ê_{n}(λ) is the set of x ∈ [−r, +r] such that the number of λ′, with ω(λ) = α(λ′), for which [x + a_{λ}^{λ′} − Δ_{1}ρ, x + a_{λ}^{λ′} + Δ_{1}ρ] ⊂ [−r, +r] − E_{n}(λ) is less than c_{3} ρ^{−(d+d′)}.

Consider first the set Ê_{n}(λ) introduced in 6.4: Ê_{n}(λ) is the set of x ∈ ℝ such that the number of λ′, with ω(λ) = α(λ′), for which [x + a_{λ}^{λ′} − Δ_{1}ρ, x + a_{λ}^{λ′} + Δ_{1}ρ] ⊂ ℝ − E_{n}(λ′) is less than (1 − τ)N_{ω(λ)} .

As N_{ω(λ)} is of the order of ρ^{−(d+d′)}, we can choose c_{3} small enough to have

c_{3} ρ^{−(d+d′)} < (1 − τ)N_{ω(λ)} (∀ λ ∈ Λ).

Now, if c_{0} > 1 is such that

c_{0}^{−1}ρ ≤ |I\underline{θ}(\underline{a})| ≤ c_{0}ρ

for all \underline{θ} ∈ Σ^{−}, \underline{a} ∈ Σ(ρ) with θ_{0} = a_{0} , we will have

|a_{λ}^{λ′}| ≤ 2 Log c_{0}

for all λ, λ′ ∈ Λ with ω(λ) = α(λ′).

Then we will have

ˆÊ_{n}(λ) ∩ [−r + 2 Log c_{0} + Δ_{1}ρ, r − 2 Log c_{0} − Δ_{1}ρ] ⊂ Ê_{n}(λ).

Now the corollary in 6.4 gives us an estimate of the desired form for ∑_{λ} p^{λ}|V_{Δρ}(Ê_{n}(λ))|,

provided

Max_{λ} |V_{Δρ}(E_{n}(λ))|

is sufficiently small.

This follows (inductively) from the desired estimate for m < n, as we have seen in 6.6, provided c̄_{1} is small enough.

We have still to control the V_{Δρ} neighbourhood of the part of ˆ Ê_{n}(λ) which lies in

[−r, −r + 2 Log c_{0} + Δ_{1}ρ] ∪ [r − 2 Log c_{0} − Δ_{1}ρ, r].

Let us consider for instance the part in [r − 2 Log c_{0} − Δ_{1}ρ, r].

The idea here is to take r large enough, to choose a subinterval J ⊂ [−r, +r] of length 5 Log c_{0} where the E_{n}(λ) are relatively thin, and to restrict to the set of λ′ which will send [−r − 2 Log c_{0} − Δ_{1}ρ, r] into J.

To be more precise, given δ > 0, we can find, if r is large enough, a subinterval J ⊂ [−r, +r] of length 5 Log c_{0} such that

∑_{λ} p^{λ}|V_{Δρ}(E_{n}(λ) ∩ J)| ≤ δ ∑_{λ} p^{λ}|V_{Δρ}(E_{n}(λ))|.

Consider now only those λ′ for which r − Log c_{0} + a_{λ}^{λ′} belongs to the interval J_{1} of same center as J and length 2 Log c_{0} ; the number of such λ’s is still of order ρ^{−(d+d′)}, in particular much larger than c_{3} ρ^{−(d+d′)} if c_{3} is small enough.

As all p^{λ} have order ρ^{(d+d′)}, we will obtain an estimate

∑_{λ} p^{λ}|V_{Δρ}( ˆÊ_{n}(λ) ∩ [r − 2 Log c_{0} − Δρ, r])| ≤ Cδ ∑_{λ} p^{λ}|V_{Δρ}(E_{n}(λ))|.

Taking δ > 0 small enough, we will have 1 > κ_{5} = 2Cδ + κ_{4} (where κ_{4} is from 6.4), which concludes the proof.

6.9 In case A) of 6.5, we just set

A = 𝔸 × 𝔸′
Λ = Σ(ρ) × Σ′(ρ)
F^{±}(\underline{a}, \underline{a}′) = \{x, ±e^{x} ∈ E(\underline{a}, \underline{a}′)\}

and deal separately with the two families (F^{+}(\underline{a}, \underline{a}′)) and (F^{−}(\underline{a}, \underline{a}′)) in the same manner than in case B).

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Carlos Gustavo T. de A. Moreira, Instituto de Matemática Pura e Aplicada – IMPA, Est. D. Castorina 110, 22460-320 Rio de Janeiro, Brasil e-mail: gugu@impa.br

J.-C. Yoccoz, Collège de France - 3, Rue d’Ulm F75005 - Paris - France