Mating parabolic rational maps with Hecke groups

Abstract

We prove that any degree (Formula presented.) 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 (Formula presented.), with the mating realised by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made by Bullett and Freiberger [Int. J. Mod. Phys. B 17 (2003), 3922–3931]. 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.