Geometry & Topology

Rotation intervals and entropy on attracting annular continua

Alejandro Passeggi, Rafael Potrie, and Martín Sambarino

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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.

Article information

Geom. Topol., Volume 22, Number 4 (2018), 2145-2186.

Received: 18 July 2016
Revised: 22 May 2017
Accepted: 1 October 2017
First available in Project Euclid: 13 April 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 37B40: Topological entropy 37B45: Continua theory in dynamics 37E45: Rotation numbers and vectors 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

rotation number entropy annular continua surface homeomorphisms horseshoes


Passeggi, Alejandro; Potrie, Rafael; Sambarino, Martín. Rotation intervals and entropy on attracting annular continua. Geom. Topol. 22 (2018), no. 4, 2145--2186. doi:10.2140/gt.2018.22.2145.

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