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Symbolic extensions and smooth dynamical systems

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Abstract

Let f:X→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 C1 non-hyperbolic (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 Cr 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∞ diffeomorphism has symbolic extensions with the same entropy. We also show that Cr generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.

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Authors and Affiliations

  1. Institute of Mathematics, Technical University of Wrocław, 50-370, Wrocław, Poland

    Tomasz Downarowicz

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824-1027, USA

    Sheldon Newhouse

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  1. Tomasz Downarowicz
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  2. Sheldon Newhouse
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Correspondence to Sheldon Newhouse.

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Downarowicz, T., Newhouse, S. Symbolic extensions and smooth dynamical systems. Invent. math. 160, 453–499 (2005). https://doi.org/10.1007/s00222-004-0413-0

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