Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products

TL;DR

This paper establishes a characterization of unique ergodicity for Anzai skew-products through the existence of an invariant conditional expectation, solving a previously open question in $C^*$-dynamical systems.

Contribution

It provides a necessary and sufficient condition for unique ergodicity of Anzai skew-products based on invariant conditional expectations.

Findings

Unique ergodicity is equivalent to the existence of a unique invariant conditional expectation.

The result answers a question posed by Abadie and Dykema in the context of $C^*$-dynamical systems.

The paper characterizes ergodic properties of Anzai skew-products in terms of operator algebraic structures.

Abstract

Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of $C^*$-dynamical systems.