Residuality of Dynamical Morphisms for Amenable Group Actions

TL;DR

This paper extends classical Baire category methods to actions of countably infinite amenable groups, showing that measures defining homomorphisms or isomorphisms are residual in certain spaces of joinings.

Contribution

It generalizes the finite generator, homomorphism, and isomorphism theorems to broader group actions using residuality arguments.

Findings

Measures defining homomorphisms form residual subsets.

Measures defining isomorphisms form residual subsets.

The approach applies to countably infinite amenable groups.

Abstract

We extend the classical Baire category approach, used in proving the finite generator theorem of Krieger, the homomorphism theorem of Sinai and the isomorphism theorem of Ornstein, applying a similar reasoning to the case of actions of countably infinite amenable groups. In principle we follow the lines of the paper by Burton, Keane and Serafin (\cite{BKS}), showing that measures defining homomorphisms or isomorphisms form residual subsets in suitably chosen spaces of joinings.