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

TL;DR

This paper proves the entropic continuity of Lyapunov exponents for certain smooth one-dimensional maps, leading to implications for entropy semi-continuity and potential integrability at high entropy measures.

Contribution

It establishes entropic continuity of Lyapunov exponents for C^r maps without assumptions on critical points, extending understanding of dynamical stability.

Findings

Lyapunov exponents are entropically continuous for large entropy maps

Entropy is upper semi-continuous at ergodic measures with large entropy

Geometric potential is uniformly integrable at high entropy measures

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.