Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension.(English)Zbl 0801.58035

Let $$f$$ be a surface diffeomorphism exhibiting a quadratic homoclinic tangency $$q$$ between the stable and unstable manifolds of a periodic saddle point $$p$$ that belongs to a basic set $$K$$ with Hausdorff dimension bigger than one. The authors prove that for almost all one-parameter families of diffeomorphisms the initial map is not a density point of hyperbolicity.
To be more precise, for a smooth family $$f_{s,t}$$ with $$f_{0,0}= f$$ and small $$| s|$$, $$| t|$$ let $$K_{s,t}$$ denote the continuation of $$K$$, and $${\mathcal F}^{\text{s}}(K_{s,t})$$ and $${\mathcal F}^{\text{u}}(K_{s,t})$$ be the stable and unstable foliations of $$K_{s,t}$$. For $$\varepsilon>0$$ small define a set $$T_{s,\varepsilon}$$ of those $$t\in (-\varepsilon,\varepsilon)$$ such that some leaf of $${\mathcal F}^{\text{u}}(K_{s,t})$$ is tangent near $$q$$ to some leaf of $${\mathcal F}^{\text{s}}(K_{s,t})$$. For each $$f_{s,t}$$, belonging to a set, defined in the article by some transversality conditions, the authors proves the following theorem: There exists $$c>0$$ such that for almost all $$s\in (-\eta,\eta)$$ with small $$\eta$$ for the Lebesgue measure $$m(\cdot)$$ one has $$\limsup_{\varepsilon\to 0}\varepsilon^{-1} m(T_{s,\varepsilon})> c$$.

MSC:

 37D99 Dynamical systems with hyperbolic behavior 37E99 Low-dimensional dynamical systems 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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 [1] [AM]Araujo, A. & Mañé, R. On the existence of hyperbolic attractors and homoclinic tangencies for surface diffeomorphisms. To appear. [2] [BC]Benedicks, M. &Carleson, L., The dynamics of the Hénon map.Ann. of Math., 133 (1991), 73–169. · Zbl 0724.58042 [3] [Bi]Birkhoff, G. D., Nouvelles recherches sur les systèmes dynamiques.Mem. Pont. Accad. Sci. Nuovi Lyncei, 1 (1935), 85–216. · JFM 60.1340.02 [4] [Bo1]Bowen, R.,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math., 470. Springer-Verlag, 1975. · Zbl 0308.28010 [5] – Hausdorff dimensions of quasi-circles.Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11–25. · Zbl 0439.30032 [6] [F]Falconer, K. J.,The Geometry of Fractal Sets. Cambridge Univ. Press, 1985. · Zbl 0587.28004 [7] [Man]Mañé, R., The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces.Bol. Soc. Brasil. Mat. (N.S.), 20 (1990), 1–24. · Zbl 0723.58029 [8] [Mar01]Marstrand, J. M., Some fundamental properties of plane sets of fractional dimensions.Proc. London Math. Soc., 4 (1954), 257–302. · Zbl 0056.05504 [9] [MM]Manning, A. &McCluskey, H., Hausdorff dimension for horseshoes.Ergodic Theory Dynamical Systems, 3 (1983), 251–261. · Zbl 0529.58022 [10] [MV]Mora, L. &Viana, M., Abundance of strange attractors.Acta Math., 171 (1993), 1–71. · Zbl 0815.58016 [11] [N]Newhouse, S., The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms.Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101–151. · Zbl 0445.58022 [12] [NP]Newhouse, S. &Palis, J., Cycles and bifurcation theory.Astérisque, 31 (1976), 44–140. · Zbl 0322.58009 [13] [P]Poincaré, H., Sur le problème de trois corps et les equations de la dynamique (Mémoire couronné du prix de S. M. le roi Oscar II de Suède).Acta Math., 13 (1890), 1–270. [14] [PT1]Palis, J. &Takens, F., Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms.Invent. Math., 82 (1985), 397–422. · Zbl 0579.58005 [15] –, Hyperbolicity and creation of homoclinic orbits.Ann. of Math., 125 (1987), 337–374. · Zbl 0641.58029 [16] [PT3]Palis, J. & Takens, F.,Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Fractal Dimensions and Infinitely Many Attractors. Cambridge Univ. Press, 1993. · Zbl 0790.58014 [17] [PV]Palis, J. & Viana, M., On the continuity of Hausdorff dimension and limit capacity for horseshoes, inDynamical Systems, Valparaiso, 1986. Lecture Notes in Math., 1331. Springer-Verlag, 1988. [18] [R]Robinson, C., Bifurcation to infinitely many sinks.Comm. Math. Phys., 90 (1983), 433–459. · Zbl 0531.58035 [19] [S]Smale, S., Diffeomorphisms with many periodic points, inDifferential and Combinatorial Topology, pp. 63–80. Princeton Univ. Press, 1965. [20] [YA]Yorke, J. A. &Alligood, K. T., Cascades of period doubling bifurcations: a prerequisite for horseshoes.Bull. Amer. Math. Soc., 9 (1983), 319–322. · Zbl 0541.58039
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