Homeomorphisms of the annulus with a transitive lift

TL;DR

This paper investigates the dynamics of orientation-preserving homeomorphisms of the annulus with transitive lifts, demonstrating the existence of a special invariant set and characterizing the rotation set as an interval containing zero.

Contribution

It establishes the existence of a specific invariant set under certain conditions and characterizes the rotation set for such homeomorphisms, extending previous understanding of annulus dynamics.

Findings

Existence of a closed, connected, invariant set with specific properties.

The rotation set of the homeomorphism is an interval containing zero.

The invariant set exhibits unbounded behavior with a negative average rotation.

Abstract

Let $f$ be a homeomorphism of the closed annulus $A$ that preserves orientation, boundary components and that has a lift $\tilde f$ to the infinite strip $\tilde A$ which is transitive. We show that, if the rotation number of both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $\Gamma\subset\tilde A$ such that $\Gamma\subset]-\infty,0]\times[0,1]$, $\Gamma$ is unlimited, the projection of $\Gamma$ to $A$ is dense, $\Gamma-(1,0)\subset\Gamma$ and $\tilde{f}(\Gamma)\subset \Gamma.$ Also, if $p_1$ is the projection in the first coordinate in $\tilde A$, then there exists $d>0$ such that, for any $\tilde z\in\Gamma,$ $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$ In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.