Continuity of entropy map for nonuniformly hyperbolic systems

TL;DR

This paper investigates the continuity properties of the entropy map in nonuniformly hyperbolic systems, showing upper semi-continuity in certain smoothness classes but not in others, challenging common assumptions.

Contribution

It establishes the upper semi-continuity of the entropy map for C1 nonuniformly hyperbolic systems with domination, contrasting with the non-continuity in C1+alpha systems.

Findings

Entropy map is upper semi-continuous for C1 systems with domination.

Entropy map is not necessarily upper semi-continuous for C1+alpha systems.

Challenges the intuition that similar systems share the same entropy map properties.

Abstract

We prove that entropy map is upper semi-continuous for C1 nonuniformly hyperbolic systems with domination, while it is not true for C1+alpha nonuniformly hyperbolic systems in general. This goes a little against a common intuition that conclusions are parallel between C1+domination systems and C1+alpha systems.