Surface homeomorphisms : the interplay between topology, geometry and dynamics

In this thesis we study certain classes of surface homeomorphisms and in particular the interplay between the topology of the underlying surface and topological, geometrical and dynamical properties of the homeomorphisms. We study three problems in three independent chapters: The first problem is to describe the minimal sets of non-resonant torus homeomorphisms, i.e. those homeomorphisms which are in a sense close to a minimal translation of the torus. We study the possible minimal sets that such a homeomorphism can admit, uniqueness of minimal sets and their relation with other limit sets. Further, we give examples of homeomorphisms to illustrate the possible dynamics. In a sense, this study is a two-dimensional analogue of H. Poincar´e’s study of orbit structures of orientation preserving circle homeomorphisms without periodic points. The second problem concerns the interplay between smoothness of surface diffeomorphisms, entropy and the existence of wandering domains. Every surface admits homeomorphisms with positive entropy that permutes a dense collection of domains that have bounded geometry. However, we show that at a certain level of differentiability it becomes impossible for a diffeomorphism of a surface to have positive entropy and permute a dense collection of domains that has bounded geometry. The third problem concerns quasiconformal homogeneity of surfaces; i.e., whether a surface admits a transitive family of quasiconformal homeomorphisms, with an upper bound on the maximal distortion of these homeomorphisms. In the setting of hyperbolic surfaces, this turns out to be a very intriguing question. Our main result states that there exists a universal lower bound on the maximal dilatation of elements of a transitive family of quasiconformal homeomorphisms on a hyperbolic surface of genus zero.