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Continuity of Lyapunov exponents for C^r one-dimensional maps

Alexandre Delplanque, Hengyi Li

TL;DR

The paper addresses how entropy and Lyapunov exponents co-vary for $\mathcal{C}^r$ interval maps with large entropy, proving that entropic convergence implies Lyapunov convergence for convergent measure sequences and establishing uniform integrability of $\log|f'|$. The authors develop a block-structure approach, decomposing dynamics into hyperbolic and neutral components, and employ a semi-local reparametrization lemma to control entropy production from neutral blocks while proving continuity for the hyperbolic part. Key contributions include a detailed entropy decomposition, a robust continuity result for the hyperbolic component, and the uniform integrability of the geometric potential, yielding corollaries on upper semi-continuity of entropy and implications for SRB measures. The results extend the understanding of entropic stability for one-dimensional maps and lay groundwork for potential generalizations to higher dimensions and broader classes of maps, even in the presence of critical points.

Abstract

We prove the entropic continuity of Lyapunov exponent for C^r maps of the interval or of the circle with large entropy for r>1, without making any assumptions on the set of critical points. A consequence is the upper semi-continuity of entropy at ergodic measures with large entropy. Another consequence is the uniform integrability of the geometric potential at ergodic measures with large entropy.

Continuity of Lyapunov exponents for C^r one-dimensional maps

TL;DR

The paper addresses how entropy and Lyapunov exponents co-vary for interval maps with large entropy, proving that entropic convergence implies Lyapunov convergence for convergent measure sequences and establishing uniform integrability of . The authors develop a block-structure approach, decomposing dynamics into hyperbolic and neutral components, and employ a semi-local reparametrization lemma to control entropy production from neutral blocks while proving continuity for the hyperbolic part. Key contributions include a detailed entropy decomposition, a robust continuity result for the hyperbolic component, and the uniform integrability of the geometric potential, yielding corollaries on upper semi-continuity of entropy and implications for SRB measures. The results extend the understanding of entropic stability for one-dimensional maps and lay groundwork for potential generalizations to higher dimensions and broader classes of maps, even in the presence of critical points.

Abstract

We prove the entropic continuity of Lyapunov exponent for C^r maps of the interval or of the circle with large entropy for r>1, without making any assumptions on the set of critical points. A consequence is the upper semi-continuity of entropy at ergodic measures with large entropy. Another consequence is the uniform integrability of the geometric potential at ergodic measures with large entropy.
Paper Structure (22 sections, 47 theorems, 161 equations)

This paper contains 22 sections, 47 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Statements of the main results
  3. Consequences for SRB measures
  4. Ideas of the proof
  5. Organization of the paper
  6. Notations
  7. Preliminary construction
  8. Hyperbolic times
  9. Entropy splitting of the hyperbolic and neutral component
  10. Defining the adapted partition
  11. Entropy of the neutral component
  12. General bounds
  13. Semi-local Reparametrization Lemma
  14. Entropy and exponent of the hyperbolic component
  15. Main Proposition
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $f : I \to I$ be a $\mathcal{C}^r$ map of positive topological entropy, with $r > 1$. Let $f_k\rightarrow f$, $\mathcal{C}^r$-weakly. Let $(\nu_k)_{k\in \mathbb{N}}$ be a sequence of $f_k$-ergodic measures, converging to some measure $\mu$ in the weak-$\ast$ topology. Then for all $\alpha>\frac{

Theorems & Definitions (86)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Corollary 1.4: Continuity of the topological entropy
  • proof
  • Definition 1.5
  • Corollary 1.6: Existence of measures of maximal entropy and entropic continuity of the Lyapunov exponent
  • proof
  • Corollary 1.7: Upper semi-continuity of the entropy at ergodic measures with large entropy
  • Theorem 1.8: Uniform integrability
  • ...and 76 more