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.

Keywords
rotation number, entropy, annular continua, surface homeomorphisms, horseshoes
Mathematical Subject Classification 2010
Primary: 37E30
Secondary: 37B40, 37B45, 37E45, 54H20