Generalized dimensions,entropies, and Liapunov exponents from the pressure function for strange sets |
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| Authors: | D. Bessis G. Paladin G. Turchetti S. Vaienti |
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| Affiliation: | (1) Service de Physique Théorique, CEN Saclay, 91191 Gif-sur-Yvette, France;(2) Laboratoire de Physique Théorique de l'École Normale Supérieure, 75231 Paris, France;(3) Laboratoire Propre du Centre National de la Recherche Scientifique, associé à l'École Normale Supérieure et à l'Université de Paris-Sud, France;(4) GNSM-CISM Unità di Roma, Italy;(5) Dipartimento di Fisica, Universitá di Bologna, I-40126 Bologna, Italy;(6) Sezione di Bologna, INFN, Italy |
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| Abstract: | For conformal mixing repellers such as Julia sets and nonlinear one-dimensional Cantor sets, we connect the pressure of a smooth transformation on the repeller with its generalized dimensions, entropies, and Liapunov exponents computed with respect to a set of equilibrium Gibbs measures. This allows us to compute the pressure by means of simple numerical algorithms. Our results are then extended to axiom-A attractors and to a nonhyperbolic invariant set of the line. In this last case, we show that a first-order phase transition appears in the pressure. |
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| Keywords: | Thermodynamic formalism topological pressure generalized Liapunov exponents generalized dimensions Renyi entropies repellers strange attractors |
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