Topological Entropy on Points without Physical-like Behaviour

TL;DR

This paper investigates the topological entropy of irregular points without physical-like behavior in certain dynamical systems, showing that these sets have full topological entropy under specific conditions.

Contribution

It establishes that in asymptotically entropy-expansive systems with the specification property, the sets of irregular points without physical-like behavior have full topological entropy.

Findings

Irregular points without physical-like behavior have full topological entropy in certain systems.

Regular points without physical-like behavior also have full topological entropy.

Results apply to a broad class of systems including Anosov diffeomorphisms and flows.

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.