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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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