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Entropy formula for surface diffeomorphisms

Yuntao Zang

TL;DR

The paper addresses the problem of expressing the topological entropy of a $C^r$ diffeomorphism on a compact surface in terms of the volume growth of tangent maps. By combining convex-geometry style decompositions of volume growth with reparametrization techniques à la Burguet–Yomdin and a careful analysis of Lyapunov exponents, it proves $h_{ m top}(f)=\lim_{n\to\infty}\frac{1}{n}\log\int_{M}\|Df^{n}_{x}\|\,dx$ under the condition $h_{ m top}(f)\geq\frac{\lambda^{+}(f)}{r}$. The approach yields a dual perspective: a global (entropy) contribution and a local (Yomdin) contribution, organized into geometric and neutral times, and culminates in an equivalence between entropy and the maximal volume growth among curves and tangent subspaces. The results generalize classical Yomdin theory to finite regularity and connect entropy with the growth of the tangent cocycle, with potential implications for computational estimation of entropy and extensions to higher dimensions. Overall, the work provides a sharp, structurally transparent formula linking dynamical complexity to derivative growth for a broad class of surface diffeomorphisms, highlighting the pivotal role of the Yomdin term and the geometry of volume growth.

Abstract

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.

Entropy formula for surface diffeomorphisms

TL;DR

The paper addresses the problem of expressing the topological entropy of a diffeomorphism on a compact surface in terms of the volume growth of tangent maps. By combining convex-geometry style decompositions of volume growth with reparametrization techniques à la Burguet–Yomdin and a careful analysis of Lyapunov exponents, it proves under the condition . The approach yields a dual perspective: a global (entropy) contribution and a local (Yomdin) contribution, organized into geometric and neutral times, and culminates in an equivalence between entropy and the maximal volume growth among curves and tangent subspaces. The results generalize classical Yomdin theory to finite regularity and connect entropy with the growth of the tangent cocycle, with potential implications for computational estimation of entropy and extensions to higher dimensions. Overall, the work provides a sharp, structurally transparent formula linking dynamical complexity to derivative growth for a broad class of surface diffeomorphisms, highlighting the pivotal role of the Yomdin term and the geometry of volume growth.

Abstract

Let be a () diffeomorphism on a compact surface with where . We establish an equivalent formula for the topological entropy: 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.
Paper Structure (18 sections, 19 theorems, 178 equations, 2 figures)

This paper contains 18 sections, 19 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Convex bounds for volume growth
  3. Mechanism behind convex bounds for volume growth
  4. Proofs of Theorem \ref{['main result']} and Theorem \ref{['entropy coincides with volume growth']}
  5. Preliminaries
  6. Lyapunov exponents
  7. Tangent dynamics
  8. Notations and conventions
  9. Volume growth of the sub-manifolds
  10. Reparametrization
  11. Decomposition of the volume growth
  12. Initial step:
  13. Inductive step:
  14. Estimation on the geometric component and proof of Theorem \ref{['main theorem of decomposition of volume growth of submanifolds']}
  15. Volume growth of the tangent cocycle
  16. ...and 3 more sections

Key Result

Theorem 1

For any $C^r$ ($r>1$) diffeomorphism $f$ on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$, we have

Figures (2)

  • Figure 1: Growth at geometric and neutral times
  • Figure 2: Local volume growth for different value of $a$

Theorems & Definitions (46)

  • Theorem 1
  • Corollary 2
  • proof
  • Theorem 3
  • Conjecture : Bur20
  • Theorem 2.1
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['main result']}
  • proof : Proof of Theorem \ref{['entropy coincides with volume growth']}
  • Proposition 4.1
  • ...and 36 more