Abstract

The aim of this series of talks is to describe some aspects of the study of the dynamics of surface homeomorphisms from the topological viewpoint. The focus will be on homeomorphisms of the torus. One of the most important concepts in this setting is the rotation set, introduced by Misiurewicz and Ziemian in 1989 as a generalization of Poincar?'s rotation number in one-dimensional dynamics. We will discuss the properties of this set, its relation with invariant measures, and the results of J. Franks relating the rational points on the rotation set with the existence of periodic orbits. We will also discuss pseudo-rotations, which are homeomorphisms with a particularly simple rotation set, and some examples of minimal pseudo-rotations with rich dynamics. The pre-requisites are kept to a minimum, and emphasis will be given to the open problems in the area.



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