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Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

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Abstract

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.

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Authors and Affiliations

  1. Departmento de Matemática, Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    Luis Barreira

  2. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, D-14195, Berlin, Germany

    Jörg Schmeling

Authors
  1. Luis Barreira
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  2. Jörg Schmeling
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Correspondence to Luis Barreira.

Additional information

Luis Barreira was partially supported by FCT’s Pluriannual Funding Program, and grants PRAXIS XXI 2/2.1/MAT/199/94, PBIC/C/MAT/2139/95, and NATO CRG970161.

Jörg Schmeling was supported by the Leopoldina-Forderpreis.

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Barreira, L., Schmeling, J. Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension. Isr. J. Math. 116, 29–70 (2000). https://doi.org/10.1007/BF02773211

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  • DOI: https://doi.org/10.1007/BF02773211

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