## 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 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 $\Phi-2(x,y)=0$.

Original language | English |
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Article number | rnx172 |

Pages (from-to) | 1921-1935 |

Number of pages | 15 |

Journal | International Mathematics Research Notices |

Volume | 2019 |

Issue number | 7 |

DOIs | |

State | Published - 1 Apr 2019 |

## ASJC Scopus subject areas

- General Mathematics