Slow entropy for abelian actions

TL;DR

This paper computes slow entropy invariants for higher rank smooth abelian actions, extending local entropy theorems and revealing universal transversal Hausdorff dimensions.

Contribution

It generalizes the Brin-Katok local entropy theorem to abelian actions and shows the universality of transversal Hausdorff dimensions.

Findings

Calculated slow entropy invariants for abelian actions in key cases.

Extended local entropy theorem to higher rank abelian actions.

Established universality of transversal Hausdorff dimensions.

Abstract

We calculate slow entropy type invariant introduced by A. Katok and J.-P. Thouvenot in [5] for higher rank smooth abelian actions for two leading cases: when the invariant measure is absolutely continuous and when it is hyperbolic. We generalize Brin-Katok local entropy Theorem to the abelian action for the above two cases. We also prove that, for abelian actions, the transversal Hausdorff dimensions are universal, i.e. dependent on the action but not on any individual element of the action.