Maps in Dimension One with Infinite Entropy

TL;DR

This paper constructs examples of one-dimensional endomorphisms with infinite topological entropy that belong to various regularity classes, including Hölder, Sobolev, and Zygmund classes, addressing open questions in the field.

Contribution

It provides explicit examples of endomorphisms with infinite entropy across multiple regularity classes, expanding understanding of dynamical systems in dimension one.

Findings

Examples of endomorphisms with infinite entropy in Hölder and Sobolev classes.

Construction of endomorphisms in Zygmund classes answering Benedicks' question.

Demonstration of the richness of dynamical behaviors within regularity constraints.

Abstract

We give examples of endomorphisms in dimension one with infinite topological entropy which are $\alpha$-H\"older and $(1,p)$-Sobolev for all $0\leq\alpha<1$ and $1\leq p<\infty$. This is constructed within a family of endomorphisms with infinite topological entropy and which traverse all $\alpha$-H\"older and $(1,p)$-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.