Actions of automorphism groups of Lie groups

TL;DR

This paper explores how subgroups of automorphism groups act on Lie groups, focusing on properties like dense orbits, invariant measures, and implications in ergodic theory and dynamical systems.

Contribution

It provides an expository overview of the structure of automorphism groups and discusses their actions in various dynamical and probabilistic contexts on Lie groups.

Findings

Analysis of actions with dense orbits

Results on invariant and quasi-invariant measures

Connections to ergodic theory and topological dynamics

Abstract

This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and quasi-invariant measures, the induced actions on the spaces of probability measures on the groups, and results concerning various issues in ergodic theory, topological dynamics, smooth dynamical systems, and probability theory on Lie groups.