Zero entropy actions of amenable groups are not dominant

2022/04/25 by Lott, Adam
#37A15 #37A20 #37A35 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.11459

Abstract

A probability measure preserving action of a discrete amenable group G is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for G = ℤ, an action is dominant if and only if it has positive entropy and that for any G, positive entropy implies dominance. In this paper, we show that the converse also holds for any G, i.e. that zero entropy implies non-dominance.

Related

• Zero entropy is generic
• Non-existence of a universal zero entropy system via generic actions of almost complete growth
• Relative uniformly positive entropy of induced amenable group actions
• Every action of a non-amenable group is the factor of a small action
• Zero-dimensional extensions of amenable group actions