#### Volume 22, issue 4 (2018)

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Rotation intervals and entropy on attracting annular continua

### Alejandro Passeggi, Rafael Potrie and Martín Sambarino

Geometry & Topology 22 (2018) 2145–2186
##### Abstract

We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a ${C}^{0}$–robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors for which the rotation interval is uniformly large but the entropy approaches zero as much as desired.

The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

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##### Keywords
rotation number, entropy, annular continua, surface homeomorphisms, horseshoes
##### Mathematical Subject Classification 2010
Primary: 37E30
Secondary: 37B40, 37B45, 37E45, 54H20
##### Publication
Received: 18 July 2016
Revised: 22 May 2017
Accepted: 1 October 2017
Published: 5 April 2018
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Jean-Pierre Otal
##### Authors
 Alejandro Passeggi Facultad de Ciencias, Centro de Matemática Universidad de la República Montevideo Uruguay Rafael Potrie Facultad de Ciencias, Centro de Matemática Universidad de la República Montevideo Uruguay Martín Sambarino Facultad de Ciencias, Centro de Matemática Universidad de la República Montevideo Uruguay