Combinatorial constructions in Smooth Ergodic Theory

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Historically motivated by problems in statistical mechanics Ergodic Theory examines statistical properties of dynamical systems. In particular, one is interested in the long-term behaviour of the system as well as the relationship between its time and space averages. One of the main questions in Ergodic Theory asks if there are smooth maps with specific ergodic properties. This is also the central question of this research project.One of the most powerful tools of constructing smooth diffeomorphisms with prescribed ergodic or topological properties is the so-called approximation by conjugation-method developed by D. Anosov and A. Katok which works on arbitrary smooth compact connected manifolds of dimension at least 2 admitting a non-trivial circle action. These diffeomorphisms are constructed as limits of conjugates of maps belonging to the circle action. In this research project we aim at the smooth realization of further ergodic as well as spectral properties. Moreover, we want to continue extending the approximation by conjugation-method to the real-analytic category.