Ergodic Theory and Dynamical Systems



Expansive algebraic actions of discrete residually finite amenable groups and their entropy


CHRISTOPHER DENINGER a1 and KLAUS SCHMIDT a2a3
a1 Mathematical Institute, Westfälische Wilhelms-University Münster, Einsteinstrasse 62, 48149 Münster, Germany (e-mail: c.deninger@math.uni-muenster.de)
a2 Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
a3 Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria (e-mail: klaus.schmidt@univie.ac.at)

Article author query
deninger c   [Google Scholar] 
schmidt k   [Google Scholar] 
 

Abstract

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma$ by automorphisms of compact abelian groups in terms of Fuglede–Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma$-action by means of a ‘fundamental homoclinic point’ and the description of entropy in terms of the renormalized logarithmic growth rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n,n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.

(Published Online April 17 2007)
(Received May 19 2006)
(Revised October 3 2006)