On the dynamics of homeomorphisms of the torus homotopic to Dehn twists.

Author(s)

Bráulio Augusto Garcia

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2012-02-02

Examining board members

Salvador Addas Zanata; Mario Jorge Dias Carneiro; Andrés Koropecki; Pedro Antonio Santoro Salomão; Fabio Armando Tal

• Advisor: Salvador Addas Zanata

Abstract

The present thesis is concerned with the dynamics of homeomorphisms of the torus homotopic to Dehn twists. We prove that if $f$ is area preserving and it has a lift $\\hat$ to the cylinder with zero flux, then either $f$ is an annulus homeomorphism, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity, for iterates of $\\hat$. This solves a version of Boyland\'s conjecture to this setting. We extend some theorems we already obtained for Dehn twists with the area preservation hypothesis to a more general class. Finally, we also give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. (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)