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Invariant sets with zero measure and full Hausdorff dimension


Authors: Luis Barreira and Jörg Schmeling
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 114-118
MSC (1991): Primary 58F15, 58F11
DOI: https://doi.org/10.1090/S1079-6762-97-00035-8
Published electronically: October 29, 1997
MathSciNet review: 1475536
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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.


References [Enhancements On Off] (What's this?)

  • 1. L. Barreira, Ya. Pesin, and J. Schmeling, On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 1, 69-72. MR 97i:58094
  • 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. Ya. Pesin and B. Pitskel', Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl. 18 (1984), no. 4, 307-318. MR 86i:28031
  • 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. CMP 97:08
  • 6. J. Schmeling and S. Troubetzkoy, Pointwise dimension for regular hyperbolic measures for endomorphisms, in preparation.

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Additional Information

Luis Barreira
Affiliation: Departamento de Matemática, Instituto Superior Técnico, 1096 Lisboa, Portugal
Email: barreira@math.ist.utl.pt

Jörg Schmeling
Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, D–14195 Berlin, Germany
Email: shmeling@math.fu-berlin.de

DOI: https://doi.org/10.1090/S1079-6762-97-00035-8
Received by editor(s): September 2, 1997
Published electronically: October 29, 1997
Additional Notes: Luis Barreira was partially supported by the projects PRAXIS XXI, 2/2.1/MAT/199/94 and JNICT, PBIC/C/MAT/2140/95. Jörg Schmeling was supported by the Leopoldina-Forderpreis.
Communicated by: Svetlana Katok
Article copyright: © Copyright 1997 American Mathematical Society

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