Statistics of rare events in infinite ergodic theory

Abstract

This thesis studies the asymptotic behaviour of the transfer operator for systems having a sigma-finite invariant measure. Firstly, a family of Markov interval maps interpolating between the Tent map and the Farey map is considered. This part considers distributional convergence of observables which are integrable, but posses singularities. It turns out that the limiting behaviour on the omega-limit set of the pole depends on the diophantine properties of the pole. Yet, distributional convergence can still be obtained on compact subsets, that do not intersect the omega-limit set of the pole and that are bounded away from the indifferent fixed point. Secondly, this thesis focuses on how to modify the transformation in general and its wandering rate in particular. It turns out that additional assumptions may be required, if the wandering rate is no longer slowly varying but regularly varying. In this part of the thesis a family of non-uniformly hyperbolic maps, known as the alpha-Farey maps is considered.