Equidistribution in measure-preserving actions of semisimple groups : case of $SL_2(\mathbb{R})$

TL;DR

This paper establishes pointwise convergence of semi-radial averages in measure-preserving actions of the group SL_2(R), extending understanding of equidistribution in this setting for functions with certain finiteness properties.

Contribution

It proves the convergence of semi-radial averages acting on K-finite L^p functions in SL_2(R) actions, a new result in the context of measure-preserving group actions.

Findings

Proves pointwise convergence of semi-radial averages for p > 1.

Extends equidistribution results to SL_2(R) actions.

Applicable to K-finite L^p functions in measure-preserving systems.

Abstract

We prove pointwise convergence for the semi-radial averages on $G=SL_2(\mathbb{R})$ given by $\int_t^{t+1} m_K\ast \delta_{a_{s}}ds$ (and similar variants), acting on $K$-finite $L^p$-functions in a probability-measure-preserving action of the group, for $p > 1$.