Some Remarks on Expanding Maps

TL;DR

This paper explores the properties of uniformly expanding maps in ergodic theory, establishing connections between regularity, conjugacies, and entropy, and introduces new rigidity results for maps on tori using differential equations.

Contribution

It presents new rigidity results for expanding maps on tori and generalizes existing theorems, employing differential equations to analyze conjugacy regularity.

Findings

Rigidity of expanding maps on the torus $\

New proof of Shub-Sullivan's Theorem on circle endomorphisms

Relations between regularity, Lyapunov exponents, and entropy for expanding maps

Abstract

In this work we treat a famous topic in Ergodic Theory and Dynamical Systems: uniformly expanding maps. We relate regularity of expanding maps and conjugacies with Lyapunov exponents, metric and topological entropies for expanding maps of the circle. In Theorem C, we present a result of rigidity of expanding maps on the torus $\mathbb{T}^2,$ after we naturally generalize it in higher dimensions. Here, we present techniques involving ordinary differential equations to study rigidity problems of expanding maps. Along of this work we provide another proof of a Shub-Sullivan's Theorem, concerning the regularity of the conjugacy between expanding endomorphism of the circle.