Slow entropy for abelian actions

TL;DR

This work extends slow entropy theory to higher-rank smooth abelian actions and derives a concrete Ledrappier-Young-type formula for $sh_\mu(\alpha,p)$ under hyperbolic or absolutely continuous invariant measures. The main result expresses slow entropy as $\sum_i \gamma_i \max_{t: p(t) \le 1} \chi_i(t)$, tying complexity to Lyapunov data and transversal dimensions, and it complements a generalized Brin-Katok local entropy theorem for actions. The authors also prove the universality of transversal Hausdorff dimensions across action elements and show two slow entropy definitions (Hamming and Bowen) coincide in the smooth setting. A reduction to a single diffeomorphism facilitates the hyperbolic and absolutely continuous analyses, linking slow entropy to rigidity phenomena and orbit equivalence considerations for higher-rank actions.

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.

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2: Slow Entropy Formula
• Theorem 1.3
• Proposition 2.1: Proposition 2.1.KKH11
• Proposition 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['THD']}