Abstract
We compute the dimension spectrumf(α) of the singularity sets of a Gibbs measure defined on a two-dimensional compact manifold and invariant with respect to aC 2 Axiom A diffeomorphism. This case is the generalization of the case where the measure studied is the Bowen-Margulis measure—the one that realizes the topological entropy. We obtain similar results; for example, the functionf 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 finally decompose these sets.
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Simpelaere, D. Dimension spectrum of axiom a diffeomorphisms. II. Gibbs measures. J Stat Phys 76, 1359–1375 (1994). https://doi.org/10.1007/BF02187066
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DOI: https://doi.org/10.1007/BF02187066