Contractive Spaces and Relatively Contractive Maps

TL;DR

This paper explores contractive spaces and relatively contractive maps, highlighting their roles as opposites to measure-preserving actions and their significance in rigidity phenomena within ergodic theory of group actions.

Contribution

It provides detailed definitions, explores their relationships with other ergodic concepts, and proves new results linking contractiveness to rigidity in group actions.

Findings

Contractive spaces are the natural opposite of measure-preserving actions.

Relatively contractive maps are the natural opposite of relatively measure-preserving maps.

Contractiveness is closely connected with rigidity phenomena in ergodic theory.

Abstract

We present an exposition of contractive spaces and of relatively contractive maps. Contractive spaces are the natural opposite of measure-preserving actions and relatively contractive maps the natural opposite of relatively measure-preserving maps. These concepts play a central role in the work of the author and J.~Peterson on the rigidity of actions of semisimple groups and their lattices and have also appeared in recent work of various other authors. We present detailed definitions and explore the relationship of these phenomena with other aspects of the ergodic theory of group actions, proving along the way several new results, with an eye towards explaining how contractiveness is intimately connected with rigidity phenomena.