Let $X$ be a set equipped with some structure (e.g. topological space, measurable space, probability space, etc.). We say that two endomorphisms $f,g \colon X \to X$ are conjugate to each other if there is an automorphism $h \colon X \to X$ such that $f = h^{-1} \circ g \circ h$; in this case, we refer to $h$ as a conjugacy from $f$ to $g$.

Now suppose the structure on $X$ naturally generates a structure of the same kind on the Cartesian product $X \times X$, with the property that for any two endomorphisms $f$ and $g$ of $X$, the direct product map $f \times g \colon (x_1,x_2) \mapsto (f(x_1),g(x_2))$ is an endomorphism of $X \times X$. We will refer to $f \times f$ as the two-point motion of $f$.

Given an endomorphism $F$ of $X \times X$, a perhaps natural question to ask is whether there exists an endomorphism $f$ of $X$ such that $F$ is conjugate to $f \times f$.

Q1. Is there a line of inquiry within dynamical systems theory and/or ergodic theory that addresses this question, or where this question plays an important role?

Now in the case that $F$ is conjugate to a two-point motion, conjugacies from $F$ to a two-point motion may be very complicated, and not very tractable to work with explicitly. In the general study of conjugacy, one is often not concerned about working explicitly with any conjugacy, but is just concerned with whether a conjugacy exists. However, one property of two-point motions that we might wish to have "explicitly" is the fact that the two components are decoupled from each other.

So we will say that an endomorphism $F$ of $X \times X$ is "nicely conjugate to a two-point motion" if there exist endomorphisms $f,g$ of $X$ that are conjugate to each other and a "tractable" (i.e. not too difficult to work with explicitly) conjugacy from $F$ to $f \times g$.

Q2. Is there a line of inquiry within dynamical systems theory and/or ergodic theory, where endomorphisms of $X \times X$ being "nicely conjugate to a two-point motion" plays an important role?