Ergodic properties of skew products in infinite measure

TL;DR

This paper investigates the ergodic properties of skew products over infinite measure spaces, establishing conditions for ergodicity and demonstrating rational ergodicity with specific limit theorems for certain classes of skew products.

Contribution

It provides new criteria for ergodicity of skew products in infinite measure spaces and analyzes their statistical properties under specific conditions.

Findings

Skew product ergodicity depends on the absence of common invariant sets among automorphisms.

Certain skew products are proven to be rationally ergodic with a  return sequence.

Trajectories satisfy central, functional central, and local limit theorems.

Abstract

Let (\Omega,\mu) be a shift of finite type with a Markov probability, and (Y,\nu) a non-atomic standard measure space. For each symbol i of the symbolic space, let \Phi_i be a measure-preserving automorphism of (Y,\nu). We study skew products of the form (\omega,y) --> (\sigma\omega,\Phi_{\omega_0}(y)), where \sigma =shift map on (\Omega,\mu). We prove that, when the skew product is conservative, it is ergodic if and only if the \Phi_i's have no common non-trivial invariant set. In the second part we study the skew product when \Omega={0,1}^Z, \mu =Bernoulli measure, and \Phi_0,\Phi_1 are R-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to \sqrt{n}, and its trajectories satisfy the central, functional central and local limit theorem.