Mathematics – Dynamical Systems
Scientific paper
2011-11-23
Mathematics
Dynamical Systems
Scientific paper
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.
Addas-Zanata Salvador
Garcia Braulio
Tal Fabio Armando
No associations
LandOfFree
Dynamics of homeomorphisms of the torus homotopic to Dehn twists does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Dynamics of homeomorphisms of the torus homotopic to Dehn twists, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dynamics of homeomorphisms of the torus homotopic to Dehn twists will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-377949