| Full text | |
| Author(s): |
Addas-Zanata, Salvador
;
Tal, Fabio A.
;
Garcia, Braulio A.
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Ergodic Theory and Dynamical Systems; v. 34, p. 14-pg., 2014-04-01. |
| 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) over cap of f to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of (f) over cap. 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) over cap to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under (f) over cap, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity. (AU) | |
| FAPESP's process: | 08/10363-5 - On the dynamics of homeomorphisms homotopic to Dehn twists |
| Grantee: | Bráulio Augusto Garcia |
| Support Opportunities: | Scholarships in Brazil - Doctorate (Direct) |