Journal of Statistical Physics

, Volume 57, Issue 1–2, pp 289–299 | Cite as

Hausdorff dimensions in two-dimensional maps and thermodynamic formalism

  • G. Paladin
  • S. Vaienti
Articles

Abstract

We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(β=dH−1)=lnj, which links the Hausdorff dimensiondH of an attractor to a free energy functionalF(β) defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.

Key words

Strange attractors thermodynamic formalism Gibbs states Hausdorff dimension generalized Lyapunov exponents 

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Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • G. Paladin
    • 1
    • 2
  • S. Vaienti
    • 3
    • 4
  1. 1.Dipartimento di FisicaUniversità “La Sapienza”RomeItaly
  2. 2.GNSM-CISM Unità di RomaRomeItaly
  3. 3.Centre Physique Théorique (Laboratoire propre du CNRS)Marseille 09France
  4. 4.Dipartimento di FisicaUniversità di BolognaBolognaItaly

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