Invariant sets with zero measure and full Hausdorff dimension

Author:

Barreira Luis,Schmeling Jörg

Abstract

For a subshift of finite type and a fixed Hölder continuous function, the zero measure invariant set of points where the Birkhoff averages do not exist is either empty or carries full Hausdorff dimension. Similar statements hold for conformal repellers and two-dimensional horseshoes, and the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously.

Publisher

American Mathematical Society (AMS)

Subject

General Mathematics

Reference6 articles.

1. On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture;Barreira, Luis;Electron. Res. Announc. Amer. Math. Soc.,1996

2. L. Barreira, Ya. Pesin, and J. Schmeling, Dimension of hyperbolic measures—a proof of the Eckmann–Ruelle conjecture, WIAS Preprint 245 and IST Preprint 26/96, 1996 (submitted for publication).

3. L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, IST Preprint 14/97, 1997 (submitted for publication).

4. Topological pressure and the variational principle for noncompact sets;Pesin, Ya. B.;Funktsional. Anal. i Prilozhen.,1984

5. Ya. Pesin and H. Weiss, A multifractal analysis of Gibbs measures for conformal expanding maps and Markov Moran geometric constructions, J. Statist. Phys. 86 (1997), no. 1/2, 233–275.

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