Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets

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.

[1]  F. Ledrappier,et al.  The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula , 1985 .

[2]  Y. Pesin.  On the notion of the dimension with respect to a dynamical system , 1984, Ergodic Theory and Dynamical Systems.

[3]  P. Walters.  INVARIANT MEASURES AND EQUILIBRIUM STATES FOR SOME MAPPINGS WHICH EXPAND DISTANCES , 1978 .

[4]  S. V. Fomin,et al.  Ergodic Theory , 1982 .

[5]  Mitchell J. Feigenbaum.  Some characterizations of strange sets , 1987 .

[6]  P. Walters.  A VARIATIONAL PRINCIPLE FOR THE PRESSURE OF CONTINUOUS TRANSFORMATIONS. , 1975 .

[7]  Intermittency in chaotic systems and Renyi entropies , 1986 .

[8]  David Ruelle.  Repellers for real analytic maps , 1982 .

[9]  Cohen,et al.  Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems. , 1985, Physical review. A, General physics.

[10]  H. Brolin.  Invariant sets under iteration of rational functions , 1965 .

[11]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[12]  M. J. Ljubich.  Entropy properties of rational endomorphisms of the Riemann sphere , 1983, Ergodic Theory and Dynamical Systems.

[13]  G. Benettin,et al.  Kolmogorov Entropy and Numerical Experiments , 1976 .

[14]  E. Ott,et al.  Dimension of Strange Attractors , 1980 .

[15]  Hirokazu Fujisaka,et al.  Statistical Dynamics Generated by Fluctuations of Local Lyapunov Exponents , 1983 .

[16]  I. P. Cornfeld.  Ergodic theory / I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai , 1982 .

[17]  L. Kadanoff,et al.  Escape from strange repellers. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Pierre Collet,et al.  The dimension spectrum of some dynamical systems , 1987 .

[19]  Anthony Manning,et al.  The dimension of the maximal measure for a polynomial map , 1984 .

[20]  A. Manning,et al.  Hausdorff dimension for horseshoes , 1983, Ergodic Theory and Dynamical Systems.

[21]  P. Grassberger.  Generalized dimensions of strange attractors , 1983 .

[22]  K. Falconer.  The geometry of fractal sets , 1985 .

[23]  M. Barnsley,et al.  Iterated function systems and the global construction of fractals , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[24]  Michał Misiurewicz,et al.  Dimension of invariant measures for maps with exponent zero , 1985, Ergodic Theory and Dynamical Systems.

[25]  Angelo Vulpiani,et al.  Characterization of strange attractors as inhomogeneous fractals , 1984 .

[26]  L. Young.  Dimension, entropy and Lyapunov exponents , 1982, Ergodic Theory and Dynamical Systems.

[27]  K. Falconer.  The geometry of fractal sets: Contents , 1985 .