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

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 C0–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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rotation number, entropy, annular continua, surface homeomorphisms, horseshoes
Mathematical Subject Classification 2010
Primary: 37E30
Secondary: 37B40, 37B45, 37E45, 54H20
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
Alejandro Passeggi
Facultad de Ciencias, Centro de Matemática
Universidad de la República
Rafael Potrie
Facultad de Ciencias, Centro de Matemática
Universidad de la República
Martín Sambarino
Facultad de Ciencias, Centro de Matemática
Universidad de la República