Definition of "Metric Conjugacy"

Question

I have a simple question, I know the definition of "Topological Conjugacy" and "Topological semi conjugacy" between two maps (Dynamical systems), but recently I have faced another kind of conjugacy that is "Metric Conjugacy" between two maps. I searched a lot on the net and in different books, but I didn't find any clear definition of it. could anyone help me know the definition of this?

Answer 1

'Metrically' normally refers to 'measure theoretically' (a kind of portmanteau) when studying ergodic theory, so searching for 'measure theoretic conjugacy' will maybe get you more hits.

In any case, a metric conjugacy between two systems $(G,X,\mu)$ and $(G,Y,\nu)$, where $G$ is a group acting by measure-preserving transformations on the probability spaces $(X,\mu)$ and $(Y,\nu)$, is a bijective measurable map $\phi \colon X' \to Y'$ such that $\phi^{-1}$ is measurable, $\phi_\ast \mu = \nu$ and $\phi(gx) = g\phi(x)$ for every $g \in G$ and $x \in X'$, where $X'$ and $Y'$ are full-measure subsets of $X$ and $Y$ respectively.