Slow entropy and differentiable models for infinite-measure preserving Z^k actions

TL;DR

This paper introduces slow entropy invariants for Z^2 actions on infinite measure spaces, revealing new limitations in realizing such actions as diffeomorphisms on compact manifolds, unlike the Z-action case.

Contribution

It develops a novel slow entropy framework for Z^2 actions and demonstrates their non-realizability as diffeomorphisms, highlighting fundamental differences from Z-actions.

Findings

Defined slow entropy invariants for Z^2 actions

Constructed Z^2 actions not realizable as diffeomorphisms

Showed differences between Z and Z^2 actions in measure-preserving realizations

Abstract

We define "slow" entropy invariants for Z^2 actions on infinite measure spaces, which measures growth of itineraries at subexponential scales. We use this to construct infinite-measure preserving Z^2 actions which cannot be realized as a group of diffeomorphisms of a compact manifold preserving a Borel measure, contrary to the situation for Z-actions, where every infinite-measure preserving action can be realized in this way.