Abstract
Some rigorous results on the dimension spectrum of expanding Markov maps of the interval are extended to Axiom AC 2 diffeomorphisms of a compact two-dimensional manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Adler and B. Weiss, Similarity of automorphisms of the torus,Mem. Am. Math. Soc. 98:1 (1969).
D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems,Russ. Math. Surv. 22:103 (1967).
V. I. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, 1968).
R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer, 1975).
P. Collet, J. Lebowitz, and A. Porzio, The dimension spectrum of some dynamical systems,J. Stat. Phys. 47:609 (1987).
J. P. Eckmann and I. Procaccia, Fluctuations of dynamical scaling indices in nonlinear systems,Phys. Rev. A 34:659 (1986).
J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).
P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, The Liapunov dimension of strange attractors,J. Diff. Equations 49:105 (1983).
P. Grassberger, R. Badii, and A. Politi, Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors,J. Stat. Phys. 51:135 (1988).
M. Gundlach, M. Sc. Dissertation, Warwick (1986).
M. Jensen, L. Kadanoff, J. Halsey, I. Procaccia, and B. Shraiman, Fractal measures and their singularities, the characterization of strange sets,Phys. Rev. A 33:1141 (1986).
J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, InLecture Notes in Mathematics, Vol. 730 (1978), p. 228.
O. Lanford,Qualitative and Statistical Theory of Dissipative Systems (CIME Lectures, 1976).
O. Lanford, InLecture Notes in Physics, Vol. 20 (1973), p. 1.
F. Ledrappier, Propriétés ergodiques des mesures de Sinai,Pub. IHES (1983), p. 163.
F. Ledrappier, Some relations between dimension and Liapunov exponents,Commun. Math. Phys. 81:229 (1981).
F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms,Ann. Math. 122:509 (1985).
Ya. B. Pesin, Characteristic Liapunov exponents and smooth ergodic theory,Russ. Math. Surv. 32:55 (1977).
Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors,Ergodic Theory Dyn. Syst. 2:417 (1982).
D. Rand, The singularities spectrumf(α) for cookie-cutters, Warwick preprint (1986).
V. A. Rholin, On the fundamental ideas of measure theory,Am. Math. Soc. Trans. 10:1 (1962).
V. A. Rholin, Exact endomorphisms of a Lebesgue space,Am. Math. Soc. Trans. 39:1 (1964).
D. Ruelle, A measure associated to Axiom A attractors,Am. J. Math. 98:619 (1976).
D. Ruelle, Thermodynamic formalism, inEncyclopedia of Mathematics, Vol. 5 (Addison-Wesley, 1978).
D. Ruelle,N. Y. Acad. Sci. 357:1 (1980).
S. Smale, Differentiable dynamical systems,Bull. Am. Math. Soc. 73:747 (1967).
Ya. G. Sinai, Gibbs measures in ergodic theory,Russ. Math. Surv. 188:21 (1972).
E. Vul, K. Khanin, and Ya. G. Sinai, Feigenbaum universality and thermodynamic formalism,Russ. Math. Surv. 39:3 (1984).
L. S. Young, Dimension, entropy and Liapunov exponents,Ergodic Theory Dyn. Syst. 2:109 (1982).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Porzio, A. The dimension spectrum of axiom a attractors. J Stat Phys 58, 923–937 (1990). https://doi.org/10.1007/BF01026557
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01026557