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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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