TOPOLOGICAL METHODS IN SURFACE DYNAMICS

This paper surveys applications of low-dimensional topology to the study of the dynamics of iterated homeomorphisms on surfaces. A unifying theme in the paper is the analysis and application of isotopy stable dynamics, i.e. dynamics that are present in the appropriate sense in every homeomorphism in an isotopy class. The first step in developing this theme is to assign coordinates to periodic orbits. These coordinates record the isotopy, homotopy, or homology class of the corresponding orbit in the suspension flow. The isotopy stable coordinates are then characterized, and it is shown that there is a map in each isotopy class that has just these periodic orbits and no others. Such maps are called dynamically minimal representatives, and they turn out to have strong global isotopy stability properties as maps. The main tool used in these results is the Thurston-Nielsen theory of isotopy classes of homeomorphisms of surfaces. This theory is outlined and then applications of isotopy stability results are given. These results are applied to the class rel a periodic orbit to reach conclusions about the complexity of the dynamics of a given homeomorphism. Another application is via dynamical partial orders, in which a periodic orbit with a given coordinate is said to dominate another when it always implies the existence of the other. Applications to rotation sets are also surveyed.