This research will focus on the theory of homeomorphisms on surfaces, specifically rotation and topological entropy. The core of the project deals with the problem of verifying the existence of essential homoclinic orbits in annulus or toral homeomorphisms by specifying that the rotational behavior be sufficiently complex, in a precise sense, as measured by the rotational entropy of the mapping. The research will also attempt to formulate a correct notion of a rotational vector on a surface of higher genus, as a means of establishing new criteria for the existence of periodic orbits of a specified rotational complexity. This research is at the interface of surface topology and dynamical systems. It makes use of interesting connections between various topological and dynamical notions in low dimensional systems.
