Rotation intervals and entropy on attracting annular continua

TL;DR

This paper investigates the relationship between rotation intervals, entropy, and attractors in annular homeomorphisms, showing conditions under which positive entropy arises and constructing examples with large rotation intervals but low entropy.

Contribution

It establishes a link between rotation intervals with multiple rotation numbers and positive entropy, and constructs examples with large rotation intervals but arbitrarily small entropy.

Findings

Positive entropy is associated with attractors having multiple rotation numbers.

Existence of $C^0$-robust rotational horseshoes in systems with positive entropy.

Examples of annular homeomorphisms with large rotation intervals but near-zero entropy.

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 so that 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.