Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms

TL;DR

This paper investigates how ergodic decompositions of Dirichlet spaces behave under order isomorphisms, establishing uniqueness and decomposability properties that deepen understanding of their structural relationships.

Contribution

It proves the uniqueness of ergodic decompositions under order isomorphisms and shows these isomorphisms decompose over ergodic components, advancing the theory of Dirichlet spaces.

Findings

Ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism.

Unitary order isomorphisms between Dirichlet spaces are decomposable over their ergodic decompositions.

The structure of Dirichlet spaces is preserved under order isomorphisms, up to conjugation.

Abstract

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.