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Communications in Mathematical Physics

, Volume 219, Issue 2, pp 443–463 | Cite as

Hausdorff Dimension of Measures¶via Poincaré Recurrence

  • L. Barreira
  • B. Saussol

Abstract:

We study the quantitative behavior of Poincaré recurrence. In particular, for an equilibrium measure on a locally maximal hyperbolic set of a C 1+α diffeomorphism f, we show that the recurrence rate to each point coincides almost everywhere with the Hausdorff dimension d of the measure, that is, inf{k>0 :f k xB(x,r)}∼r d . This result is a non-trivial generalization of work of Boshernitzan concerning the quantitative behavior of recurrence, and is a dimensional version of work of Ornstein and Weiss for the entropy. We stress that our approach uses different techniques. Furthermore, our results motivate the introduction of a new method to compute the Hausdorff dimension of measures.

Keywords

Entropy Recurrence Rate Hausdorff Dimension Equilibrium Measure Dimensional Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • L. Barreira
    • 1
  • B. Saussol
    • 2
  1. 1.Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal.¶E-mail: luis.barreira@math.ist.utl.ptPT
  2. 2.LAMFA-CNRS FRE 2270, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France. E-mail: benoit.saussol@u-picardie.frFR

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