Mating parabolic rational maps with Hecke groups

2024/07/20 by Bullett, Shaun, Lomonaco, Luna, Lyubich, Mikhail +1 · 3 citations
#30C10 #30F10 #30F35 #37C85 (Secondary) #37F05 #37F10 #37F31 #37F32 (Primary) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2407.14780

Abstract

We prove that any degree d rational map having a parabolic fixed point of multiplier 1 with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group H_d+1, with the mating realized by an algebraic correspondence. This solves the parabolic version of the Bullett-Freiberger Conjecture from 2003 on mateability between rational maps and Hecke groups. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering.

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