Entropy formula for surface diffeomorphisms

TL;DR

This paper derives a new formula for the topological entropy of surface diffeomorphisms, linking it to the integral of the norm of derivatives and volume growth, with applications to Lyapunov exponents.

Contribution

It establishes an equivalent entropy formula involving the integral of the derivative norm and characterizes entropy via volume growth, extending recent Lyapunov exponent studies.

Findings

Derived an entropy formula: $h_{top}(f)=\lim_{n\to\infty}\frac{1}{n}\log\int_{M}\|Df^{n}_{x}\|dx.

Connected topological entropy to volume growth of curves.

Applied the formula to analyze Lyapunov exponents and entropy continuity.

Abstract

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$ where $\lambda^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an equivalent formula for the topological entropy: $$h_{\rm top}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int_{M}\left\|Df^{n}_{x}\right\|\,dx.$$ We also characterize the topological entropy via the volume growth of curves and several applications are presented. Our approach builds on the key ideas developed in the works of Buzzi-Crovisier-Sarig (\emph{Invent. Math.}, 2022) and Burguet (\emph{Ann. Henri Poincar\'e}, 2024) concerning the continuity of the Lyapunov exponents.