Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting

TL;DR

This paper proves the upper semi-continuity of metric entropy for certain diffeomorphisms with dominated splitting, under conditions on Lyapunov exponents, advancing understanding of entropy behavior in dynamical systems.

Contribution

It establishes the upper semi-continuity of the entropy map for $C^{r}$ diffeomorphisms with dominated splitting under specific Lyapunov exponent conditions, a novel result in dynamical systems theory.

Findings

Upper semi-continuity of entropy map proven for measures with positive and non-positive Lyapunov exponents.

Results apply to sequences of invariant measures and individual measures with dominated splitting.

Advances understanding of entropy stability in partially hyperbolic systems.

Abstract

For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the entropy map in the following two cases: (1) if a sequence of invariant measures has only positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the upper limit of their metric entropies is less than or equal to the entropy of the limiting measure; (2) if an invariant measure has positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the entropy map is upper semi-continuous at this measure.