Topological Entropy on Points without Physical-like Behaviour

Abstract

We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one maps of transitive Anosov flows. We consider the nonempty set of physical-like measures that attracts the empirical probabilities (i.e. the time averages) of Lebesgue-almost all the orbits. We define the set $I_f\cap {\Gamma }_f\subset M$ of irregular points without physical-like behaviour. We prove that, if not all the invariant measures of $f$ satisfy Pesin Entropy Formula (for instance in the Anosov case), then $I_f\cap {\Gamma }_f$ has full topological entropy. We also obtain this result for some class of asymptotically entropy-expansive continuous maps on a compact metric space, if the set of physical-like measures are equilibrium states with respect to some continuous potential. Finally, we prove that also the set $(M\setminus I_f)\cap {\Gamma }_f$ of regular points without physical-like behaviour, has full topological entropy.

15 pages

References in corpus (2)

• Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting
• Symbolic extensions for partially hyperbolic diffeomorphisms

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