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 $\tilde{A}=V\circ U$ 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 Latt`es map. For flexible Latt`es maps $L=L_j$ induced by the multiplication by 2 on elliptic curves with given $j$-invariant we provide a very precise description of $[L]$. Namely, we show that any rational function equivalent to $L_j$ necessarily has the form $L_{j^{\mathrm{\prime }}}$ for some $j^{\mathrm{\prime }}\in \mathbb{C}$, and that the set of $j’\in \mathbb C$suchthat$ L_{j’}\sim L_{j}$coincideswiththeorbitof$j$ under the correspondence associated with the classical modular equation ${\mathrm{\Phi }}_2(x,y)=0$.