Mating quadratic maps with the modular group II

Abstract

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences F a :

Correspondences in Complex Dynamics

“…Bullett and Penrose [1] conjectured that for every a ∈ C Γ , the correspondence F a is a mating between some quadratic map f c (z) = z 2 + c and the modular group PSL(2, Z). More recently, this conjecture was settled affirmatively by Bullett and Lomonaco [16], provided the quadratic family is replaced by a quadratic family of parabolic maps (see figures 4 and 5).…”

“…We require our fundamental domains to be simply-connected and bounded by Jordan curves (see Figure 3). In [16] we show that {a ∈ C : |a − 4| ≤ 3, a = 1} ⊂ K, and that when a is in the interior of this disk the standard fundamental domains (see figure 3) are a Klein combination pair. More generally we prove that for every a ∈ K, we can always choose a Klein combination pair whose boundaries ∂∆ Cov and ∂∆ J are transversal to the attracting-repelling axis at P .…”

“…In [16], using the theory of parabolic-like maps developed by the second author (see [29]), the first two authors proved the following (see figures 4 and 5):…”

“…defined by the polynomial equation (4). As part of the proof of Theorem 2.1 it is shown in [16] that: Theorem 2.2 (Böttcher map) If a ∈ C Γ , there is a unique conformal homemorphism ϕ a : Ω a → H such that…”

Quadrature domains and the real quadratic family

“…We also remark that a notion of conformal matings between Kleinian groups and polynomials as holomorphic correspondences was introduced in [BP94] (see also [BL20]). The definition of conformal mating used in the present work follows more closely that of [LLMM18a].…”