Abstract
We compute the dimension spectrumf(α) of the singularity sets of the Bowen-Margulis measure defined on a two-dimensional compact manifold and invariant with respect to aC 2 Axiom A diffeomorphism. It is proved thatf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we decompose these sets.
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References
R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer, Berlin, 1973).
P. Collet, J. L. Lebowitz and A. Porzio, Dimension spectrum of some dynamical systems,J. Stat. Phys. 47:609–644 (1987).
P. Collet and F. Koukiou, Large deviations for multiplicative chaos,Commun. Math. Phys. 147:329–342 (1992)
M. Denker, C. Grillenberger, and K. Sigmund,Ergodic Theory on Compact Spaces (Springer, Berlin, 1976).
R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics (Springer-Verlag, Berlin, 1985).
J. Franchi, Chaos multiplicatif: Un traitement simple et complet de la fonction de partition, preprint (1993).
T. C. Hasley, M. H. Jensen, L. P. Kadanoff, I. Proccacia, and B. I. Shraiman, Fractal measures and their singularities: The characterization of strange sets,Phys. Rev. B 33: 1141–1151 (1986).
F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques,Z. Wahr. Verw. Geb. 30:185–202 (1974).
F. Ledrappier, Some relations between dimension and Lyapunov exponents,Commun. Math. Phys. 81:229–238 (1981).
F. Ledrappier, On the dimension of some graphs,Contemp. Math 135:285–293 (1992).
F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension,Ann Math. 122:540–574 (1985).
A. O. Lopes, Entropy and large deviations,Nonlinearity 3:527–546 (1990).
R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces,Bol. Soc. Bras. Mat. 20(2):1–24 (1990).
A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy,Ergod. Theory Dynam. Syst. 5:109–124 (1981).
A. Manning and H. McCluskey, Hausdorff dimensions for horseshoes,Ergod. Theory Dynam. Syst. 5:251–260 (1983).
A. Porzio, The dimension spectrum of Axiom A attractors,J. Stat. Phys. 58:923–937 (1990).
D. A. Rand, The singularity spectrumf(α) for cookie-cutters,Ergod. Theory. Dynam. Syst. 9:527–541 (1989).
D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading, Massachusetts 1978).
D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms,Topology 14:319–327 (1975).
Y. G. Sinai, Markov partitions andC-diffeomorphisms,Funkt. Anal. Ego Prilozheniya 2(1):64–89 (1968).
D. Simpelaere, Dimension spectrum of Axiom A diffeomorphisms. II: Gibbs measures,J. Stat. Phys., this issue.
D. Simpelaere, Multifractal decomposition of the Sierpinski carpet, preprint (1993).
L. S. Young, Dimension, entropy and Lyapunov exponents,Ergod. Theory Dynam. Syst. 2:109–124 (1982).
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Simpelaere, D. Dimension spectrum of axiom a diffeomorphisms. I. The Bowen-Margulis measure. J Stat Phys 76, 1329–1358 (1994). https://doi.org/10.1007/BF02187065
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DOI: https://doi.org/10.1007/BF02187065