Entropy formula for surface diffeomorphisms

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{λ^{+}(f)}{r}$ where $λ^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left|Df^{n}{x}\right|$. We establish an equivalent formula for the topological entropy: $$h{\rm top}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int_{M}\left|Df^{n}_{x}\right|,dx.$$ 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.

💡 Research Summary

The paper by Yuntao Zang establishes a precise relationship between the topological entropy of a $C^{r}$ ($r>1$) diffeomorphism on a compact surface and the volume growth of its tangent cocycle. The main hypothesis is that the topological entropy $h_{\rm top}(f)$ dominates the “Yomdin term’’ $\lambda^{+}(f)/r$, where $\lambda^{+}(f)=\lim_{n\to\infty}\frac1n\max_{x\in M}\log|Df^{n}_{x}|$ is the maximal Lyapunov exponent in the $C^{1}$ norm. Under this condition the author proves two central theorems.

Theorem A states that the entropy can be recovered from a purely topological quantity:
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