Semi-stable laws for intermittent maps

Abstract

In this thesis we study probabilistic limit theorems for one-dimensional non-uniformly expanding maps with a single neutral fixed point, commonly known as intermittent maps. In 2004, S. Gouëzel showed that generic Hölder observables satisfy a stable law under the dynamics of the Liverani-Saussol-Vaienti (L.S.V.) family of intermittent maps in the case that an absolutely continuous probability measure is preserved. A key reason for the appearance of stable laws in the setting of Gouëzel’s result is the fact that the return time to a particular reference set is regularly varying. We investigate what occurs when this regular variation is not present. In particular, we consider modifications of the L.S.V. map where stable laws fail to hold for generic Hölder observables and show that instead semi-stable laws emerge. We further establish that these semi-stable laws also appear in the context of the usual L.S.V. map for a certain class of oscillatory observables.