Recomposing rational functions

Let $A$ be a rational function. A rational function $\tilde A$ is called an elementary transformation of $A$ if there exist rational functions $U$ and $V$ such that $A = U \circ V$ and $\tilde A = V \circ U$. We say that rational functions $A$ and $B$ are equivalent, and write $A \sim B$, if there exists a chain of elementary transformations between $A$ and $B$. The equivalence class $[A]$ of $A$ is a union of conjugacy classes, and the relation $A \sim B$ can be considered as a weakened form of the classical conjugacy relation.
In the talk, we provide conditions for the finiteness of the number of conjugacy classes in $[A]$, and reduce two important problems about rational functions to describing $[A]$. The first problem is the problem of describing rational functions commuting with a given function $A$. The second one is the problem of describing “dynamical symmetries” of a given function $A$, where by dynamical symmetries we mean the group of Mobius transformations $\mu$ such that $A^{\circ k} \circ \mu = A^{\circ k}$ for some $k \ge 1$.