For any two ergodic measure-preserving transformations on nonatomic probability spaces, there is an isomorphism between the two probability spaces carrying orbits onto orbits.
Dye's Theorem
See also
Ergodic TheoryExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Dye's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DyesTheorem.html