On multidimensional infinite dihedral group extensions of Gibbs Markov maps

TL;DR

This paper establishes a local central limit theorem for cocycles in multidimensional infinite dihedral group extensions of Gibbs Markov maps, revealing their ergodic or dissipative behavior based on dimension.

Contribution

It introduces a novel approach using irreducible representations to analyze non-abelian, non-compact group extensions where convolution methods fail.

Findings

Derived local central limit theorem for these group extensions.

Identified conditions for mixing and dissipativity based on dimension.

Provided asymptotics for the first return time to the origin.

Abstract

We obtain a local central limit theorem for cocycles associated with a class of non abelian and non compact group extensions of Gibbs Markov maps. This class consists of multidimensional infinite dihedral groups. Unlike in the set up of the random walks on groups, we cannot use the convolution of measures on the group and instead we resort to an approach based on irreducible representations. Depending on the dimension of the group, we obtain either mixing, and thus ergodicity, or dissipativity. Also, we obtain the asymptotics of the first return time of the group extension to the origin.