In this thesis, we investigate three different phenomena in uniformly hyperbolic dynamics. First, we study entry time statistics for -mixing actions. More specifically, given a -mixing dynamical system (X ,T, BX,µ) we find conditions on a family of sets {Hn ⇢ X : n 2 N} so that µ(Hn)⌧n tends in law to an exponential random variable, where ⌧n is the entry time to Hn. We apply this to hyperbolic toral automorphisms, and we obtain that µ(Hn)⌧n tends in law to an exponential random variable when {Hn ⇢ X : n 2 N} are shrinking sets along the unstable direction. Second, we prove escape rate results for special flows over subshifts of finite type, over conformal repellers and over Axiom A diffeomorphisms. Finally, we study escape rates for Axiom A flows. Our results are based on a discretisation of the flow and the application of the results in [39]. Third, we study the smoothness of the stationary measure with respect to smooth perturbations of the iterated function scheme and the weight functions that define it. Our main theorems relate the smoothness of the perturbation of: the iterated function scheme and the weight functions; to the smoothness of the perturbation of the stationary measure. The results depend on the smoothness of: the iterated function scheme and the weights functions; and the space on which the stationary measure acts as a linear operator.