Entropy formula for surface diffeomorphisms

Abstract

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\mathrm{top}}(f)\ge \frac{\hat{I}{»}^{+}(f)}{r}$ where $\hat{I}{»}^{+}(f):={\lim }_{n\to +\infty }\frac{1}{n}{\max }_{x\in M}\log \Vert D{f}_x^n\Vert$. We establish an equivalent formula for the topological entropy: $$h_{\mathrm{top}}(f)=\underset{n\to +\infty }{\lim }\frac{1}{n}\log {\int }_M\Vert D{f}_x^n\Vert dx.$$ 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 Poincaré}, 2024) concerning the continuity of the Lyapunov exponents.

errors corrected, new applications and further questions added, 56 pages, 3 figures