Occupation times of sets of infinite measure for ergodic transformations

Abstract

Assume that T is a conservative ergodic measure-preserving transformation of the infinite measure space $(X,\mathcal{A},\mu)$. We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling–Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with continued-fraction (CF)-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These abstract results are illustrated by applications to interval maps with indifferent fixed points.