Slow Entropy for Abelian Actions

Nome: 
Changguang Dong - Penn State University, USA.
Instituição: 
Universidade da Pensilvânia
Data do Evento: 
terça-feira, 19 de Dezembro de 2017 - 14:00
Local do evento
Sala 221
Descrição: 

Metric entropy is an important numerical invariant in dynamical systems. It reflects exponential orbit growth rate of a system in measure theoretic sense, which is well studied in smooth ergodic theory for Z and R actions. However, for "large" group actions, direct extension of metric entropy fails. Therefore one may try to find some other entropy type invariants. In this talk, we calculate slow entropy type invariant introduced by A. Katok and J.-P. Thouvenot for higher rank smooth abelian actions for two leading cases: when the invariant measure is absolutely continuous and when it is hyperbolic. As a by-product, we generalize Brin-Katok local entropy Theorem to the abelian action for the above two cases. We will 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. We will start with basic definitions of entropy.