Communications in Mathematical Physics

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

Hausdorff Dimension of Measures¶via Poincaré Recurrence

  • L. Barreira
  • B. Saussol


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.


Entropy Recurrence Rate Hausdorff Dimension Equilibrium Measure Dimensional Version 
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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:
  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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