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Topological entropy of level sets of empirical measures for non-uniformly expanding maps

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  • In this article we obtain a variational principle for saturated sets for maps with some non-uniform specification properties. More precisely, we prove that the topological entropy of saturated sets coincides with the smallest measure theoretical entropy among the invariant measures in the accumulation set. Using this fact we provide lower bounds for the topological entropy of the irregular set and the level sets in the multifractal analysis of Birkhoff averages for continuous observables. The topological entropy estimates use as tool a non-uniform specification property on topologically large sets, which we prove to hold for open classes of non-uniformly expanding maps. In particular we prove some multifractal analysis results for C1-open classes of non-uniformly expanding local diffeomorphisms and Viana maps [1,33].

    Mathematics Subject Classification: Primary: 37D35, 37D20; Secondary: 37D25, 37C50.

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