Recomposing rational functions

2016/10/20 by Pakovich, Fedor · 3 citations
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1610.06411

Abstract

Let A be a rational function. For any decomposition of A into a composition of rational functions A=U∘ V the rational function \widetilde A=V∘ 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 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'∈ \mathbb C, and that the set of j'∈ \mathbb C such that L_j'∼ L_j coincides with the orbit of j under the correspondence associated with the classical modular equation Φ₂(x,y)=0.

Cited by

• On the multiplier spectrum of polynomials
• Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture
• On intertwined polynomials

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