Dynamics of homeomorphisms of the torus homotopic to Dehn twists

TL;DR

This paper investigates the dynamics of torus homeomorphisms homotopic to Dehn twists, establishing conditions for bounded motion, the presence of rotation intervals, and implications for topological entropy and Boyland's conjecture.

Contribution

It provides new criteria linking vertical rotation sets to boundedness and entropy, and proves a version of Boyland's conjecture for area-preserving cases.

Findings

Zero vertical rotation set implies existence of an invariant horizontal set.

Explicit condition for the rotation set to contain an interval.

In area-preserving case, either all points are bounded or there are points with positive and negative vertical velocities.

Abstract

In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under $f$. In other words, if we consider the lift $\hat{f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat{f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If $f$ is area preserving and has a lift $\hat{f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat{f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.