Continuity properties of partial entropy

TL;DR

This paper provides a criterion for the upper semi-continuity of partial entropy in $C^{1+ ext{alpha}}$ diffeomorphisms, linking it to the continuity of Lyapunov exponents and extending previous results.

Contribution

It establishes a general criterion for entropic semi-continuity based on Lyapunov exponents, applicable in all dimensions and for various dynamical systems.

Findings

Entropy is upper semi-continuous at generic ergodic measures.

The criterion applies to measures with dominated splittings and SRB measures.

Extends $C^{ ext{infinity}}$ results to $C^{1+ ext{alpha}}$ diffeomorphisms.

Abstract

We establish a general criterion on the upper semi-continuity of partial entropy in all directions for $C^{1+\alpha}$ diffeomorphisms: it holds when the respective sums of Lyapunov exponents are continuous. This addresses, in arbitrary dimensions, the converse aspect of the entropic continuity of the Lyapunov exponents established by Buzzi, Crovisier, and Sarig. Consequently, the entropy (and all the partial entropies) is always upper semi-continuous at generic ergodic measures of every $C^{1+\alpha}$ diffeomorphism, which extends the $C^{\infty}$ result of Newhouse. Numerous applications and examples are provided, including topics related to measures with dominated splittings, SRB measures, average expanding diffeomorphisms, singular flows, standard maps, and symbolic codings for diffeomorphisms.