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Cantor sets, dynamics and arithmetic. (Conjuntos de Cantor, dinâmica e aritmética.) (Portuguese) Zbl 0965.11032

22\(^o\) Colóquio Brasileiro de Matemática. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA). iv, 77 p. (1999).
The booklet under review surveys and outlines the main ideas behind the proofs of some recent results by the author (partly co-authored by J.-C. Yoccoz), some of them still unpublished, some others published in [Ann. Inst. Henri Poincaré Anal. Non Linéaire 13, 747–781 (1996; Zbl 0865.58035)].
While these results are concerned with two completely different mathematical areas (homoclinic bifurcations for diffeomorphisms on surfaces, on the one hand, and continuous fractions, on the other hand), they are linked by the notion of regular Cantor set. Let \(I_1,\ldots, I_k\) be pairwise disjoint compact subintervals of \(\mathbb{R}\) and let \(I\) be their convex hull. Assume that \(\psi:I_1\cup\cdots \cup I_k\rightarrow I\) is an expanding function of the class \(C^q\) for some real number \(q>1\), that is, a \(C^1\) map satisfying \(|\psi'(x)|>1\) for any \(x\) and for which there is a constant \(C>0\) with \(|\psi'(x)-\psi'(y)|\leq C|x-y|^{q-1}\) for any \(x,y\). Further, assume that, for any \(j\), \(\psi(I_j)\) is the convex hull of the union of some of the intervals \(I_i\), and that \(\psi^n(I_{j,n})= I_1\cup\cdots \cup I_k\) for any number \(n\) large enough, where \(I_{j,n}\subset I_j\) is the set on which \(\psi^n\) is well defined. Then we call the Cantor set \(K\) of all points from \(I\) on which \(\psi^n\) is well defined for any \(n\) a regular Cantor set of the class \(C^q\).
Regarding regular Cantor sets, the paper includes, among others, the following results:
Theorem 1. (jointly with J.-C. Yoccoz) Fix \(1<q\leq \infty\) and let \(\Omega\) be the set of pairs \((K_1,K_2)\) of regular Cantor sets of the class \(C^q\) such that the sum of their Hausdorff dimensions \(HD(K_1)+HD(K_2)\) is greater than \(1\). Then there is an open dense set \(\mathcal U\subset \Omega\) in the \(C^q\)-topology such that, if \((K_1,K_2)\in \mathcal U\) and \(S(K_1,K_2)\subset \mathbb{R}\) is the set of numbers \(t\) with the property that any pair \((\widetilde{K}_1,\widetilde{K}_2)\) close enough to \((K_1,K_2+t)\) satisfies \(\widetilde{K}_1\cap\widetilde{K}_2\neq \emptyset\), then \(S(K_1,K_2)\) is open and dense in \(K_1-K_2\).
In the above theorem we mean \(K_2+t=\{y+t: y\in K_2\}\) and \(K_1-K_2=\{x-y:x\in K_1,y\in K_2\}\). In the \(C^q\)-topology, two Cantor sets \(K\) and \(\widetilde{K}\) are close when the corresponding families of intervals \(I_1,\ldots,I_k\) and \(\widetilde{I}_1,\ldots,\widetilde{I}_k\) in their definitions have the same cardinality, the endpoints of the intervals \(I_i\) are close to those of the intervals \(\widetilde{I}_i\), the maps \(\psi\) and \(\widetilde{\psi}\) are close in the \(C^1\)-topology, and the constants \(C\) and \(\widetilde{C}\) are close.
Theorem 2. If \(K_1\) and \(K_2\) are Gauss regular Cantor sets then \(HD(K_1+K_2)=HD(K_1)+HD(K_2)\).
Here, so-called Gauss regular Cantor sets are those obtained by restricting the Gauss map \(g:(0,1]\rightarrow [0,1)\) defined by \(g(x)=1/x-[1/x]\) to a finite numbers of intervals having as their endpoints irrationals of the type \(r+\sqrt{s}\), with \(r,s\in \mathbb{Q}\).
The above theorems have a number of consequences. Theorem 1, which proves a conjecture by J. Palis from the eighties, implies the following remarkable result:
Theorem 3. Let \(\{\varphi_\mu\}_{\mu\in \mathbb{R}}\) a family of \(C^2\)-diffeomorphisms on a closed surface \(M^2\) having a homoclinic \(\Omega\)-explosion in \(\mu=0\). Then, generically (in the \(C^2\)-topology for diffeomorphisms on \(M^2\)), the set of parameters \(\mu\) such that \(\varphi_\mu\) is either persistently hyperbolic or has stable homoclinic tangencies, has Lebesgue full density in \(\mu=0\).
Here a diffeomorphism \(\varphi\) is hyperbolic if its nonwandering set \(\Omega(\varphi)\) is hyperbolic, and has a homoclinic tangency if \(\varphi\) has a hyperbolic saddle fixed point \(p\) whose stable and unstable manifolds \(W^s(p)\) and \(W^u(p)\) intersect tangentially at some point \(x\); “persistence” and “stability” means that the properties above persist under small perturbations of \(\varphi\). Finally, \(\{\varphi_\mu\}_{\mu\in \mathbb{R}}\) has a homoclinic \(\Omega\)-explosion in \(\mu=0\) if, roughly speaking, \(\varphi_u\) is persistently hyperbolic for any \(\mu<0\) and \(\varphi_0\) has a homoclinic tangency as before; moreover \(\Omega(\varphi_0)=\Lambda\cup \mathcal O\), where \(\Lambda\) is hyperbolic and \(\mathcal O=\{\varphi^n(x)\}_{n=-\infty}^\infty\) is the set of (quadratic) homoclinic tangencies between \(W^s(p)\) and \(W^u(p)\).
Theorem 2 can be applied to number theory as follows. For any \(\alpha\in \mathbb{R}\), let \( k(\alpha)=\sup \{k>0: \text{}|\alpha-p/q|<1/(kq^2)\) has infinitely many rational solutions \(p/q\)

MSC:

11J06 Markov and Lagrange spectra and generalizations
28A78 Hausdorff and packing measures
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11K60 Diophantine approximation in probabilistic number theory

Citations:

Zbl 0865.58035
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