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April 2020 Maps in dimension one with infinite entropy
Peter Hazard
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Ark. Mat. 58(1): 95-119 (April 2020). DOI: 10.4310/ARKIV.2020.v58.n1.a7

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.

Funding Statement

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.

Citation

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Peter Hazard. "Maps in dimension one with infinite entropy." Ark. Mat. 58 (1) 95 - 119, April 2020. https://doi.org/10.4310/ARKIV.2020.v58.n1.a7

Information

Received: 3 November 2017; Revised: 13 September 2019; Published: April 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ARKIV.2020.v58.n1.a7

Subjects:
Primary: 37B40
Secondary: 26A16, 37E05, 46E35

Rights: Copyright © 2020 Institut Mittag-Leffler

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Vol.58 • No. 1 • April 2020
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