Arkiv för Matematik

Volume 58 (2020)

Number 1

Maps in dimension one with infinite entropy

Pages: 94 – 119

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n1.a7

Author

Peter Hazard (Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, Brazil)

Abstract

For each real $\alpha , 0 \leq \alpha \lt 1$, we give examples of endomorphisms in dimension one with infinite topological entropy which are $\alpha$‑Hölder; and for each real $p , 1 \leq p\lt \infty$, we also give examples of endomorphisms in dimension one with infinite topological entropy which are $(1, p)$-Sobolev. These examples are constructed within a family of endomorphisms with infinite topological entropy and which traverse all $\alpha$-Hölder 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.

Keywords

entropy, Hölder classes, Sobolev classes, Zygmund classes

2010 Mathematics Subject Classification

Primary 37B40. Secondary 26A16, 37E05, 46E35.

This work has been partially supported by “Projeto Temático Dinâmica em Baixas Dimensões” FAPESP Grant 2011/16265-2, by FAPESP Grant 2015/17909-7, and by CAPES Projeto PVE CNPq 401020/2014-2.

Received 3 November 2017

Received revised 13 September 2019

Accepted 30 September 2019

Published 21 July 2022