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Entropy Continuity of Lyapunov Exponents for Non-flat 1-dimensional Maps

Hengyi Li

Abstract

We show that the continuity property of Lyapunov exponents proved in \cite{BCS-Exponents} for smooth surface diffeomorphisms extends to smooth interval maps, in the case when the map only has non-flat critical points and the entropies converging to the topological entropy. The result we obtained is stronger than the continuity of Lyapunov exponents. In particular, we prove the uniform integrability of Lyapunov exponents over entropies.

Entropy Continuity of Lyapunov Exponents for Non-flat 1-dimensional Maps

Abstract

We show that the continuity property of Lyapunov exponents proved in \cite{BCS-Exponents} for smooth surface diffeomorphisms extends to smooth interval maps, in the case when the map only has non-flat critical points and the entropies converging to the topological entropy. The result we obtained is stronger than the continuity of Lyapunov exponents. In particular, we prove the uniform integrability of Lyapunov exponents over entropies.

Paper Structure

This paper contains 12 sections, 16 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Statement of our main result
  3. Strategy: entropy and uniform integrability defect
  4. Organization of the paper
  5. Notations
  6. Shadowing Intervals
  7. Basic Properties
  8. Inductive Construction of $\mathcal{S}(x)$
  9. Proof of Theorem 2.1
  10. The Entropy Gap
  11. Conditions for Entropy Gap
  12. Proof of the Main Theorem

Key Result

Theorem 1.1

Let $f:X\to X$ be a smooth map with only non-flat critical points and let $(\mu_k)_{k\in \mathbb{N}}$ be a sequence of probability measures on $X$ approximating $f$ in entropy. Denote its weak limit by $\mu$. If $h_{\rm top}(f)>0$, then

Theorems & Definitions (38)

  • Definition 1
  • Theorem 1.1
  • Definition 2
  • Lemma 1.2
  • Theorem 1.3
  • Definition 3
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 4
  • ...and 28 more