A note on quasi-similarity of Koopman operators

TL;DR

This paper constructs examples of ergodic automorphisms with Markov quasi-similar Koopman operators, addressing a question by A. Vershik and exploring invariants of this quasi-similarity.

Contribution

It provides the first known examples of non-weakly isomorphic ergodic automorphisms with Markov quasi-similar Koopman representations and discusses related metric invariants.

Findings

Constructed two ergodic automorphisms with Markov quasi-similar Koopman operators

Addressed a question of A. Vershik regarding quasi-similarity

Discussed metric invariants of Markov quasi-similarity

Abstract

Answering a question of A. Vershik we construct two non-weakly isomorphic ergodic automorphisms for which the associated unitary (Koopman) representations are Markov quasi-similar. We also discuss metric invariants of Markov quasi-similarity in the class of ergodic automorphisms.