I am interested in isomorphisms between transformations with measures of them, especially is the transformations that are not invertible. I found the following definition:
Two measure-preserving transformations $(X,\,\mathbb{B},\,\mu,\,f)$ and $(Y,\,\mathbb{A},\,\nu,\,g)$ are called isomorphic if there exist measure-preserving mappings $\phi:X\to Y$, $\psi:Y\to X$ such that the following conditions hold: 1. $\psi\circ\phi=id\;$ $\mu$-almost everywhere, 2. $\phi\circ\psi=id\;$ $\nu$-almost everywhere, 3. $\phi\circ f= g\circ \phi\;$ $\mu$-almost everywhere, 4. $\psi\circ g= f\circ \psi\;$ $\nu$-almost everywhere.
Is it right that, if X=Y, instead we can just use the following conditions: 3. $\phi\circ f= g\;$ $\mu$-almost everywhere, 4. $\psi\circ g= f\;$ $\nu$-almost everywhere?
More interestingly, is it possible to have an isomorphism between two mappings one which is non-invertible? For example, the doubling map or the tent map $f(x)=1-|1-2x|$, $0\leq x<1$ on a circle both preserve several measures, including the Lebesgue measure, but are not invertible. Is it possible to prove or disprove them to be isomorphic to an interval exchange transformation? To a rotation of a circle? To each other?
Thank you for answering me.