Dynamics of homeomorphisms of the torus homotopic to Dehn twists

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 f of f to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of 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 f to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under f , or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion

“…F : R → R is a lift of an orientation preserving homeomorphism of T 1 = R/Z and its rotation number is 0, then |F n (x) − x| is uniformly bounded (a property known as uniformly bounded deviations). A similar property was proved in [AGT11] for homeomorphisms of T 2 in the homotopy class of a Dehn twist (x, y) → (x + y, x).…”

Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior

“…In the homotopic to the identity case, given a vector (cos(θ), sin(θ)), we define . In lemma 2 of [5] we proved that, if (0, 0) ∈ ρ( f ) or 0 ∈ ρ V ( f ), then B ∞ θ , B ∞ S and B ∞ N are non-empty, closed subsets of the plane, positively invariant under f . It is easy to see that their omega-limits satisfy the following (because…”

A characterization of annularity for area-preserving toral homeomorphisms

“…The result from [Sli02] was generalized in [AGT11], where the technical and restrictive twist condition was replaced by the condition that the homeomorphism of T 2 be in a Dehn homotopy class (which implies a weak type of topological twist).…”