Hausdorff dimensions in two-dimensional maps and thermodynamic formalism |
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Authors: | G. Paladin S. Vaienti |
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Affiliation: | (1) Dipartimento di Fisica, Università La Sapienza, I-00185 Rome, Italy;(2) GNSM-CISM Unità di Roma, Rome, Italy;(3) Centre Physique Théorique (Laboratoire propre du CNRS), Luminy case 907, F-13288 Marseille 09, France;(4) Dipartimento di Fisica, Università di Bologna, Bologna, Italy |
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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(=dH–1)=lnj, which links the Hausdorff dimensiondH of an attractor to a free energy functionalF() defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map. |
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Keywords: | Strange attractors thermodynamic formalism Gibbs states Hausdorff dimension generalized Lyapunov exponents |
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