Arkiv för Matematik
Volume 58 (2020)
Maps in dimension one with infinite entropy
Pages: 94 – 119
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.
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