Recomposing Rational Functions

Let $A$ be a rational function. For any decomposition of $A$ into a composition of rational functions $A=U\circ V$ the rational function $\widetilde A=V\circ U$ is called an elementary transformation of $A$, and rational functions $A$ and $B$ are called equivalent if there exists a chain of elementary transformations between $A$ and $B$. This equivalence relation naturally appears in the complex dynamics as a part of the problem of describing of semiconjugate rational functions. In this paper we show that for a rational function $A$ its equivalence class $[A]$ contains infinitely many conjugacy classes if and only if $A$ is a flexible Lattès map. For flexible Lattès maps $\mathcal L=\mathcal L-j$ induced by the multiplication by 2 on elliptic curves with given $j$-invariant we provide a very precise description of $[\mathcal L]$. Namely, we show that any rational function equivalent to $\mathcal L-j$ necessarily has the form $\mathcal L-{j'}$ for some $j'\in {\mathbb C}$, and that the set of $j'\in {\mathbb C}$ such that $\mathcal L-{j'}\sim \mathcal L-{j}$ coincides with the orbit of $j$ under the correspondence associated with the classical modular equation $\Phi-2(x,y)=0$.