Singular potentials, rigidity and recurrence in low dimensional dynamics

In the first part of this thesis we study the behaviour of the equilibrium measure and the Birkhoff sums for a singular potential over the doubling map. A complete multifractal analysis for the the Birkhoff sums and the equilibrium measure (for the ’appropriate’ scaling) is given in Paper I. In Paper II we prove a parameter continuity property of the pressure function for a family of singular potentials. In the third paper we study some properties of the invariant sets of a general expanding Markov map of the circle and investigate a rigidity related question, proving that for a fixed such map there are not many (in the topological sense) other maps so that they share common compact invariant sets of ’small’ Hausdorff dimension. The last project deviates from the previous ones and focuses on recurrence properties of hyperbolic systems and specifically, hyperbolic automorphisms of the two-dimensional torus. For such a system, we provide a formula for the Hausdorff dimension of the so-called uniform recurrence set.