Knowlpedia

Definition

Measurable conjugacy

An isomorphism of probability spaces intertwining their transformations.

A measurable conjugacy between probability-preserving systems (X,μ,T)(X,\mu,T) and (Y,ν,S)(Y,\nu,S) is an b:X→Yb:X\to Y such that

b∘T=S∘balmost everywhere.b\circ T=S\circ b\quad\text{almost everywhere}.

Thus S=bTb−1S=bTb^{-1} wherever the representatives are defined.

Consequences

Conjugacy preserves ergodicity, mixing, and distributions of time averages. Pullback Wf=f∘bWf=f\circ b is a unitary L2(Y,ν)→L2(X,μ)L^2(Y,\nu)\to L^2(X,\mu) intertwining the forward Koopman operators. The converse implication from an arbitrary unitary intertwiner to measurable conjugacy is generally false: the function-algebra structure carries additional information.