On the loss of upper semi-continuity of metric entropy for $C^{r}$ diffeomorphisms

TL;DR

This paper provides an optimized upper bound estimate for the loss of upper semi-continuity of metric entropy in $C^r$ diffeomorphisms, demonstrating sharpness through examples.

Contribution

It introduces a refined estimate for entropy loss in $C^r$ diffeomorphisms, improving previous bounds and confirming their sharpness.

Findings

Established an upper bound estimate for entropy loss.

Optimized the estimate considering dimension and Lipschitz constant.

Proved the estimate is sharp using Buzzi's examples.

Abstract

In this article, we give an upper bound estimate for the quantitative loss of the upper semi-continuity of the metric entropy for $C^r\:(r>1)$ diffeomorphisms. Building on earlier entropy estimates and reparametrization methods, we optimize the upper bound estimate with respect to both dimension and asymptotic Lipschitz constant. Motivated by Buzzi's examples, we show that the estimate is sharp.