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On the continuity of Hausdorff dimension and limit capacity for horseshoes

Part of the Lecture Notes in Mathematics book series (LNM,volume 1331)

Abstract

We consider Hausdorff dimension and limit capacity of basic sets (horse - shoes) of C1 two-dimensional diffeomorphisms and show that they depend continuously on the diffeomorphism. For the restriction of the horseshoe to a stable (unstable) manifold, the result had been proved by McCluskey and Manning using the thermodyna - mic formalism. Our proof simply makes use of Hölder conjugancies between nearby horseshoes and Hölder stable and unstable foliations with Hölder exponents close to one. As a consequence, the local Hausdorff dimension and limit capacity at any point of the horseshoe are equal and independent of the point. Also, for an open and dense subset of C1 surface diffeomorphisms their horseshoes have Hausdorff dimension smaller than two.

Keywords

  • Limit Capacity
  • Hausdorff Dimension
  • Stable Manifold
  • Stable Foliation
  • Unstable Foliation

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© 1988 Springer-Verlag

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Palis, J., Viana, M. (1988). On the continuity of Hausdorff dimension and limit capacity for horseshoes. In: Bamón, R., Labarca, R., Palis, J. (eds) Dynamical Systems Valparaiso 1986. Lecture Notes in Mathematics, vol 1331. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083071

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  • DOI: https://doi.org/10.1007/BFb0083071

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50016-2

  • Online ISBN: 978-3-540-45889-0

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