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\def\Om{\Omega}
\def\ve{\varepsilon}
\def\re{\Bbb R}
\def\a{\alpha}
\def\be{\beta}
\def\La{\Lambda}
\def\teu{\underline\theta}
\def\Z{\Bbb Z}
\def\N{\Bbb N}
\def\Q{\Bbb Q}
\def\vr{\varphi}
\def\au{\underline\alpha}
\def\beu{\underline\beta}
\def\la{\lambda}
\def\L{\Cal L}

\centerline{\bigbf Sums of regular Cantor sets, dynamics and}
\centerline{\bigbf applications to number theory}

\vglue .5in

\centerline{Carlos Gustavo T. de A. Moreira}

\vglue .2in

\centerline{\bf I M P A}

\centerline{\sl Estrada Dona Catorina 110}
\centerline{\sl Jardim Botanico}
\centerline{\sl CEP 22460-320}
\centerline{\sl Rio de Janeiro-RJ, Brazil}
\centerline{\sl e-mail: gugu\@impa.br}

\vglue .4in

\flushpar
{\bf 1. \,\, Introduction and main results}

\vglue .1in


Let $K_1,K_2$ be regular $(C^{1+\ve})$ Cantor sets. We say that $K_1$ has {\it stable intersection\/} with $K_2$ if for any pairs of regular Cantor sets $(\tilde K_1,\tilde K_2)$ near $(K_1,K_2)$ (in the $C^{1+\ve}$ topology) we have $\tilde K_1\cap\tilde K_2\ne\emptyset$.

We say that $K_1$ has $d$-stable intersection with $K_2$ $(d>0)$ if for any $(\tilde K_1,\tilde K_2)$ near $(K_1,K_2)$, we have $HD(\tilde K_1\cap\tilde K_2)\ge d$.

Clearly, $\{t\in\re \mid K_1 \text{ has stable intersection with } K_2+t\}$ is contained in the interior of $\{t\in\re\mid K_1\cap (K_2+t)\ne\emptyset\}=
\{x-y,\,\, x\in K_1,\,\, y\in K_2\}=K_1-K_2$.

The following theorem gives a positive answer to a conjecture by J. Palis (this is a joint work with J.C. Yoccoz).

\medskip

\flushpar
{\bf Theorem:} Let $\Om=\{(K_1,K_2)$ regular Cantor sets, s.t. $HD(K_1)+HD(K_2)>1\}$. There is an (open and) dense set $U\subset\Om$ in the $C^k$ topology $(1<k\le \infty)$ such that
$$
(K_1,K_2)\in U\Rightarrow S(K_1,K_2):=\{t\in\re\mid K_1 \text{ has stable intersection with } K_2+t\}
$$
is (open and) dense in  $K_1-K_2$ and $HD((K_1-K_2)\setminus S(K_1,K_2))<1$.

Moreover, given $(K_1,K_2)\in\Om$ and $0<d<HD(K_1)+HD(K_2)-1$, we can approximate $(K_1,K_2)$ by $(\tilde K_1,\tilde K_2)$ such that $S_d(K_1,K_2)=\{t\in\re\mid K_1$ has $d$-stable intersection with  $K_2\}$ satisfies the above properties


\vglue .3in

\flushpar
{\bf 2.\,\, Application to dynamics}

\vglue .2in

Consider a first homoclinic bifurcation associated to a hyperbolic set $\La$ (a family of diffeomorphisms $\vr_\mu\colon M^2 \to M^2$, $\mu\in(-1,1)$ such that:

$\bullet$ $\vr_\mu$ is persistently hyperbolic for $\mu<0$;

$\bullet$ $\vr_0$ presents a homoclinic tangency associated to the periodic saddle point $p_0$

\medskip

Palis and Takens proved that if $HD(\La)=HD(K^s)+HD(K^u)<1$ then $0$ is a point of full density of hyperbolicity in the parameter line.

The ideas of our previous theorem allow us to prove that with open and dense conditions, if $HD(\La)>1$, then there is an open set of parameters $\mu$, with positive density at $0$, corresponding to diffeomorphisms presenting stable homoclinic tangencies.

(This phenomena may persistentely coexist with positive density of hyperbolicity for positive parameters $\mu$ at $\mu=0$).

\vglue .5in

\flushpar
{\bf 3. \,\, Applications to number theory}

\vglue .2in

\flushpar
{\bf Theorem:} If $K_1$ and $K_2$ are regular Cantor sets, $K_1$ is of Class $C^2$ and not essentially affine then
$$
HD(K_1+K_2)=\min\{1,HD(K_1)+HD(K_2)\}.
$$
This is a consequence of the main lemma in the proof of the previous theorem.

This is the main tool in the proof of the following results about the Markov and Lagrange spectra:

Let $\a\in\re\setminus\Q$. We define $k(\a)=\sup\{k>0\mid |\a-\frac pq| <\frac1{kq^2}$ have infinitely many solutions $\frac pq\in\Q\}$. The Lagrange spectrum $L$ is the set $\{k(\a),\a\in\re\setminus\Q, k(\a)<\infty\}$.

The Markov spectrum can be defined as $M=\{\dsize\inf_{(x,y)\in\Z^2\setminus(0,0)}| f(x,y)|)^{-1}$, $f(x,y)=ax^2+bxy+cy$, $a,b,c\in\re$, $b^2-4ac=1\}$.

It can be proved that $L$ and $M$ are closed sets and $L\subset M$.

Markov proved that $L\cap(-\infty,3)=M\cap(-\infty,3)=\{k_1=\sqrt5<k_2=2\sqrt2< k_3=\frac{\sqrt{221}}5 < \dots\}$ where $(k_n)$ is a sequence of algebraic numbers converging to $3$.

M. Hall proved that $L$ contains a whole half-line (for instance $[6,\infty)$), as a consequence of the fact that $K+K$ is an interval, where $K$ is the (regular) Cantor set of the numbers in $[0,1]$ whose continued fractions consist of 1,2,3 and 4.

\vglue .3in

\flushpar
{\bf Theorem:}
\flushpar
$\bullet$ If $d(t):=HD(L\cap(-\infty,t))$, then $d(t)$ is also equal to $HD(M\cap(-\infty,t))$, and is a continuous surjective function from $\re$ to $[0,1]$.

Moreover,

i) $d(t)=\min\{1,2\cdot HD(k^{-1}(-\infty,t))\}$

ii) $\max\{t\in\re\mid d(t)=0\}=3$

iii) $d(\sqrt{12})=1$.

\medskip

\flushpar
$\bullet$ The set of accumulation points of $L$ is perfect $(L'=L'')$

\flushpar
$\bullet$ $0<HD(M\setminus L)<1$.

If the continued fraction of $\a$ is $[a_0,a_1,a_2,\dots]$ then we have \linebreak $k(\a)=\dsize\limsup_{n\to\infty}(\a_n+\be_n)$, where
$$
\a_n:=[a_n,a_{n+1},a_{n+2},\dots]
$$
and
$$
\be_n:=[0,a_{n-1}, a_{n-2},\dots,a_1].
$$

Let $\Sigma=\N^{\Z}$ be the set of all bi-infinite sequences of positive integers. If $\teu=(a_n)_{n\in\Z}\in\Sigma$, define $f(\teu)=\a_0+\be_0=[a_0,a_1,a_2,\dots]+[0;a_{-1},a_{-2},\dots]$.
We have $L=\{\dsize\limsup_{n\to\infty}f(\sigma^n\teu), \teu\in\Sigma\}$ and
$M=\{\sup_{n\in\Z}f(\sigma^n\teu), \teu\in\Sigma\}$, where $\sigma\colon\Sigma\to\Sigma$ is the shift defined by $\sigma((a_n)_{n\in\Z})=(a_{n+1})_{n\in\Z}$.

From this is clear that $L\subset M$, and the relation between sums of Cantor sets of continued functions and the geometric structures of $L$ and $M$.

If $\vr\colon M^2\to M^2$ is a diffeomorphism, $\La$ is a hyperbolic set for $\vr$ and $f\colon M^2\to\re$ is a differentiable function we may define sets that are analogous to the classical Markov and Lagrange spectra, the dynamical spectra
$$
L(\La,f)=\{\dsize\limsup_{n\to\infty}f(\vr^n(x)), x\in\La\}
$$
$$
M(\La,f)=\{\sup_{n\in\Z}f(\vr^n(x)), x\in\La\}.
$$

We have results analogous of those of the previous theorem for $L(\La,f)$ and $M(\La,f)$ that hold generically in an open set of $(\vr,f)$ that contains an open and dense set of $\{(\vr,f)$ such that $\vr$ is conservative\}, exept for the last one \linebreak $(0<HD(M\setminus L)<1)$, that may be false. In this open set it is also generically true that if $d_{\La,f}(t)=HD(L(\La,f)\cap(-\infty,t))$ then
$\inf\{t\in\re\mid d_{\La,f}(t)=1\}$ belongs to the closure of the interior of
$L(\La,f)$.

\vglue .2in

\flushpar
{\bf Sketch of the proof of the main theorem:}

\bigskip

Roughly speaking, a regular Cantor set $k$ in the real line is defined by an iterated function system of contractions (say $f_1,f_2,\dots,f_n)$ whose images are disjoint that satisfies some combinatorial conditions (related to the existence of a Markov partition). Some compositions $f_i\circ f_j$ may be not defined, so let $B=\{(i,j),\,\, 1\le i, \,\, j\le n$ such that $f_i\circ f_j$ is defined\}. Let $\Sigma^-=\{\teu=(\theta_k)_{k\le 0}$, $\theta_{k}\in (1,2,\dots,n\}$ and $(\theta_k,\theta_{k+1})\in B, \,\, \forall k\}$.

For $\teu=(\dots,\theta_{-k},\dots,\theta_{-1},\theta_0)\in\Sigma^-$ and $m\in\N$ we define $\teu^m=(\theta_{-m},\theta_{1-m},\dots,\mathbreak \theta_{-1},\theta_0)$ and $f_{\teu^m}=f_{\theta_{-m}}\circ f_{\theta_{1-m}}\circ\dots\circ f_{\theta_0}$.
Let $B(\teu^m)$ be the affine map such that $B(\teu^m)\circ f_{\teu^m}$ is orientation preserving and have the interval $[0,1]$ as image. It is a well-know result that for each $\teu\in\Sigma^-$ there exists $k_{\teu}=\dsize\lim_{m\to\infty} (B_{\teu^m}\circ f_{\teu^m})$ and the $k_{\teu}$ depend H\"older-continuously in $\teu$ with the metric $d(\teu,\tilde{\teu})=|I(\teu\cap\tilde{\teu})|$, where
$\teu\wedge\tilde{\teu}=(\theta_{-k},\dots,\theta_{-1},\theta_0)$, $\theta_i=\tilde\theta_i$ for $i\ge-k$ and $\theta_{-k-1}\ne\tilde\theta_{-k-1}$, and $I(\teu\wedge\tilde{\teu})$ is the image of $f_{\teu^k}$ (the dependence on $\teu$ is even Lipschtz if the maps $f_i$ are of class $C^2$).

The image of $K$ by the maps $k_{\teu}$ are the {\it limit geometries\/} of $K$. They are also regular Cantor sets, and intervals of the construction of a limit geometry intersected to the limit geometry are small copies of another limit geometry. This allows us to define renormalization operators as follows:

Let $K,K'$ be two regular Cantor sets, $K_{\teu}=k_{\teu}(K)$ and $K_{\teu'}'=k_{\teu'}'(K')$ be two limit geometries of $K$ and $K'$, $\au$ and $\beu$ be two finite sequences, $\la\in\re^*$ and $t\in\re$. We define the renormalization operator $R_{\au,\beu}$ by $R_{\au,\beu}\colon \Sigma^-\times\Sigma^-\times\re^*\times\re$, $R_{\au,\beu}(\teu,\teu',\la,t)=
(\teu\au,\teu'\beu,\la(d-c)/(b-a), (\la c+t-a)/(b-a))$
where $[a,b]$ is the image of $I_{\au}$ by $K_{\teu}$ and $[c,d]$ is the image of $I_{\beu}'$ by $k_{\teu'}'$.

A point $(\teu,\teu',\la,t)\in\Sigma^{'-}\times\Sigma^-\times\re^*\times\re$ can be viewed as a relative position of two limit geometries
($K_{\teu}$ and $\la K_{\teu'}'+t$). $R_{\au,\beu}(\teu,\teu',\la,t)$ is the relative position of two intervals of the construction of $K_{\teu}$ and
$\la K_{\teu'}'+t$ (that are $k_{\teu}(I_{\au})$ and $\la k_{\teu'}'(I_{\beu}')+t)$.

The strategy of the proof of the main theorem is based on the following proposition (that is not difficult to prove):

\bigskip

\flushpar
{\bf Proof:} If there is a compact set $\L\subset\Sigma^-\times\Sigma^-\times\re^*\times\re$ with non-empty interior such that for each $(\teu,\teu',\la,t)\in\L$ there is a renormalization operator $R_{\au,\beu}$ such that $R_{\au,\beu}(\teu,\teu',\la,t)\in\text{int }\L$ then $K_{\teu}$ has stable intersection with $\la K_{\teu'}'+t$ for any $(\teu,\teu',\la,t)\in\L$.

Such a set $\L$ is called a compact recurrent set.

In the previous proposition, if we require that for each $(\teu,\teu',\la,t)\in\L$ there are renormalization operators $R_{\au_i,\beu_i}$, $1\le i\le j=j(\teu,\teu',\la,t)$ such that $\dsize\sum_{i=1}^j\la_i^d>1$, where $\la_i=|k_{\teu}(I_{\au_i})|$ then $K_{\teu}$ has $d$-stable interseciton with
$\la K_{\teu'}'+t$ for any $(\teu,\teu',\la,t)\in\L$.

Such a set $\L$ is called a compact $d$-recurrent set.

We prove that we can always perturb a pair of regular Cantor sets $(K,K')$ such that $HD(K)+HD(K')>1$ in order to have a compact set as in the previous proposition.

A theorem by Marstrand shows that under our hypothesis, for any $(\teu,\teu')\in\Sigma^-\times\Sigma^{'-}$ there is a set with positive measure of $(\la,t)\in\re^*\times\re$ for which $K_{\teu}\cap(\la K_{\teu'}'+t)$ has positive Hausdorff dimension. For $(\teu,\teu',\la,t)$ in this set there are many candidates for renormalization operators $R_{\au,\beu}$ (when some point of $K_{\teu}\cap(\la K_{\teu'}'+t)$ belongs to $k_{\teu}(I_{\au})\cap k_{\teu'}'(I_{\beu}'))$ that can return to the same set, and small neighbourhoods
of this set give reasonable candidates for the set $\L$ in the proposition, and some of these sets $\L$ will work after most of small random perturbations of the functions that define $K$.

The main lemma that we use in the proof is the Scale recurrence lemma, that shows that, given a set $L\subset\Sigma^-\times\Sigma^{'-}\times\re^*$ such that for some large number $A>0$ the measure of $\{\la\in[-A,A]\mid (\teu,\teu',\la)\notin L\}$ is small for any $(\teu,\teu')$ there exists a set $F$ contained in a small neighbourhood of $L$ such that the measure of $\{\la\in[-A,A]\mid (\teu,\teu',\la)\notin F\}$ is small for any $(\teu,\teu')$ that is compact recurrent set at the level of $\Sigma^-\times\Sigma^{'-}\times\re^*$ (we forget the coordinate $t$), and indeed a 
$d$-recurrent set with $d$ very close to $HD(K)+HD(K')$ at the level of $\Sigma^-\times\Sigma^{'-}\times\re^*$.

The Scale recurrence lemma holds with the hypothesis of the set $K$ being essentially non-affine. This means that there are $\teu,\tilde{\teu}\in\Sigma^-$ and $x\in K_{\tilde{\teu}}$ such that $(k_{\teu}\circ k_{\tilde{\teu}}^{-1})''(x)\ne0$. It is applied to the set given by Marstrand's theorem, and it is necessary since the renormalization operators are very expansive in respect to the coordinate $t$, but essentially neutral in the coordinate $\la$, so we use the lemma to obtain a compact recurrent set at the level of $(\teu,\teu',\la)$ and then use perturbations to obtain a recurrent compact set in $\Sigma^-\times\Sigma^{'-}\times\re^*\times\re$.

The Scale recurrence lemma can be used to show the formula of the dimension of $K+K'$ that we used in the discussion about the Markov and Lagrange Spectra.

In the references below there are some papers related to the results discussed here:

\vglue .5in

\centerline{\bf REFERENCES}

\vglue .1in

\item{[BPMV]} R. Bam\'on, S. Plaza, C.G. Moreira and J. Vera, \ Differentiable structures of central Cantor sets, Ergod. Th. and Dynam. Syst. 17 (1997), pp. 
1027--1042.

\item{[BPV]} R. Bam\'on, S. Plaza and J. Vera, \ On Central Cantor Sets with 
self-arithmetic difference of positive Lebesgue measure, J. London Math. Soc. 51 (1995), pp. 137--146.

\item{[CF]} T.W. Cusick and M.E. Flahive,\ The Markoff and Lagrange spectra, Math. Surveys and Monographs, no. 30, A.M.S. (1989).

\item{[F]} G.A. Freiman,\ Diophantine approximation and geometry of numbers (The Markoff spectrum), Kalininskii Gosudarstvennyi Universitet, Moscow, 1975.

\item{[H]} M. Hall, \ On the sum and product of continued fractions, Annals of Math., Vol. 48, (1947), pp. 966--993.

\item{[L]} Larsson, Per, \ L'ensemble difference de deux ensembles de Cantor aleatoires, C.R. Acad. Sci. Paris S\'er. I Math. 310 (1990), no. 10, 735--738.

\item{[M1]} C.G. Moreira, \ Stable intersections of Cantor sets and homoclinic bifurcations, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 13 (1996), no. 6, pp. 741--781.

\item{[M2]} C.G. Moreira, \ Geometric properties of the Markov and Lagrange spectra. To appear.

\item{[MO]} P. Mendes and F. Oliveira, \ On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity, Vol. 7, (1994), pp. 329--343.

\item{[MY1]} C.G. Moreira and J.-C. Yoccoz, \ Stable intersections of regular Cantor sets with large Hausdorff dimensions. Preprint - IMPA, 1998.

\item{[MY2]} C.G. Moreira and J.-C. Yoccoz, \ Tangences homoclines stables pour les ensembles hyperboliques de grande dimension fractale. To appear.

\item{[N1]} S. Newhouse, \ Non density of Axiom $A(a)$ on $S^2$, Proc. A.M.S. Symp. Pure Math., Vol. 14, (1970), pp. 191--202.

\item{[N2]} S. Newhouse, \ Diffeomorphisms with infinitely many sinks, Topology, Vol. 13, (1974), pp. 9--18.

\item{[N3]} S. Newhouse, \ The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHES, Vol. 50, (1979), pp. 101--151.

\item{[P]} J. Palis, \ Homoclinic orbits, hyperbolic dynamics and fractional dimension of Cantor sets, Contemporary Mathematics 58, (1987), pp. 203--216.

\item{[P1]} J. Palis, \ Homoclinic bifurcations, sensitive chaotic dynamics and strange attractors, Dynamical Syst. and Related Topics. World Scientific, (1991), pp. 466-473.

\item{[PT]} J. Palis and F. Takens, \ Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms, Invent. Math., Vol. 82, (1985), pp. 379--442.

\item{[PT1]} J. Palis and F. Takens, \ Hyperbolicity and the creation of homoclinic orbits, Annals of Math., Vol. 125, (1987), pp. 337--374.

\item{[PT2]} J. Palis and F. Takens, \ Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractors, Cambridge Univ. Press, 1992.

\item{[PY]} J. Palis and J.C. Yoccoz, \ Homoclinic Tangencies for Hyperbolic sets of large Hausdorff Dimension Bifurcations, Acta Mathematica, Vol. 172, (1994), pp. 91--136.

\item{[PY1]} J. Palis and J.C. Yoccoz, \ On the arithmetic sum of regular Cantor sets, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 14  (1997), pp. 439--456.

\item{[S]} A. Sannami, \ An example of a regular Cantor set whose difference set is a Cantor set with positive measure, Hokkaido Math. Journal, Vol. XXI (I), (1992), pp. 7--23.

\item{[Su]} D. Sullivan, \ Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 15--223, Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, RI, (1988).


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