Prime spectrum and dynamics for nilpotent Cantor actions

Pacific Journal of Mathematics, vol. 327, pp. 107–128

Abstract

A minimal equicontinuous action by homeomorphisms of a discrete group on a Cantor set X is locally quasianalytic if each homeomorphism has a unique extension from small open sets to open sets of uniform diameter on X.A minimal action is stable if the action on X of the closure of in the group of homeomorphisms of X is locally quasianalytic.When is virtually nilpotent, we say that : × X → X is a nilpotent Cantor action.We show that a nilpotent Cantor action with finite prime spectrum must be stable.We also prove there exist uncountably many distinct Cantor actions of the Heisenberg group, necessarily with infinite prime spectrum, which are not stable.

Authors 2

1. Steven Hurder

   University of Illinois Chicago

   Affiliation as printed

   Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, IL, United States

2. Olga Lukina Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, Leiden, Netherlands