Maps in dimension one with infinite entropy

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.

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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