On the ergodic theory of certain non-discrete actions and topological orbit equivalences

TL;DR

This paper investigates quasi-invariant measures for non-discrete diffeomorphism groups with Morse-Smale dynamics, extending circle results to higher dimensions and showing that continuous orbit equivalences are almost everywhere diffeomorphisms.

Contribution

It extends ergodic theory results from circle actions to higher-dimensional manifolds for groups containing Morse-Smale dynamics, and characterizes orbit equivalences as diffeomorphisms.

Findings

Extension of circle group results to higher dimensions

Continuous orbit equivalences coincide with diffeomorphisms almost everywhere

Identification of quasi-invariant measures for specific group actions

Abstract

Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. These results are also applied to show that, for many groups as above, every continuous orbit equivalence must coincide almost everywhere with a diffeomorphism of the corresponding manifold.