Dynamics on surfaces

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In dynamical systems theory the understanding of low-dimensional systems is of majorimportance, since this often allows to elucidate and describe basic mechanisms and paradigmatic examples that prove to be relevant in a much broader context. While the understanding of one-dimensional systems is fairly complete and one of the great success stories of the field, two-dimensional systems are far less well-understood and fundamental problems in this area are still wide open. The aim of the project is to make use of recent developments and elaborate new tools that have become available in the last years in order to address a number of central problems in surface dynamics. The main focus lies on the following three topics.1. Classification of zero entropy systems on surfaces.2. Transition to chaos in surface dynamics.3. Rotation theory in dimension twoMore specifically, item 1 aims at a generalisation of a recent classification of area-preserving C-infinity diffeomorphisms of the sphere with zero entropy by Franks and Handel. Item 2 addresses a conjecture by C. Tresser from 1983 on the occurrence of period doubling cascades on the boundary of chaos in surface dynamics. Item 3 includes the study of the remaining cases of the Franks-Misiurewicz conjecture, stated in 1991, concerning the non-existence of certain types of aperiodic dynamics and the related rotation sets on the two-torus.