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Symbolic extensions and smooth dynamical systems. (English) Zbl 1067.37018
Summary: Let \(f : X \rightarrow X\) be a homeomorphism of the compact metric space \(X\). A symbolic extension of \((f,X)\) is a subshift on a finite alphabet \((g,Y)\) which has \(f\) as a topological factor. We show that a generic \(C^{1}\) nonhyperbolic (i.e., non-Anosov) area-preserving diffeomorphism of a compact surface has no symbolic extensions. For \(r>1\), we exhibit a residual subset \(\mathcal R\) of an open set \(\mathcal U\) of \(C^r\) diffeomorphisms of a compact surface such that if \(f\in\mathcal R\), then any possible symbolic extension has topological entropy strictly larger than that of \(f\). These results complement the known fact that any \(C^\infty\) diffeomorphism has symbolic extensions with the same entropy. We also show that \(C^r\) generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.

MSC:
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37B10 Symbolic dynamics
37B40 Topological entropy
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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