Recomposing rational functions

TL;DR

This paper characterizes the equivalence classes of rational functions under elementary transformations, showing that only flexible Lattès maps have infinitely many conjugacy classes within their equivalence class, and describes these classes explicitly.

Contribution

It proves that only flexible Lattès maps have infinitely many conjugacy classes in their equivalence class and provides a detailed description of these classes for Lattès maps induced by multiplication on elliptic curves.

Findings

Equivalence class of a rational function contains infinitely many conjugacy classes iff it is a flexible Lattès map.

For Lattès maps, the equivalence class corresponds to the orbit under a modular correspondence.

Provides a precise description of the set of rational functions equivalent to a given Lattès map.

Abstract

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 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'}$ for some $j'\in \mathbb C$, and that the set of $j'\in \mathbb C$ such that $ L_{j'}\sim L_{j}$ coincides with the orbit of $j$ under the correspondence associated with the classical modular equation $\Phi_2(x,y)=0$.