Schwarz reflections and anti-holomorphic correspondences

Abstract

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in [LLMM18a,LLMM18b]. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic rational maps with the modular group. Contents 34 7. Tessellation of The Escape Locus 40 8. Properties of The Straightening Map 9. Homeomorphism between Model Spaces 49 10. The Associated Correspondences Appendix A. A Family of Parabolic Anti-rational Maps 58 References 66

Quadrature Domains and the Real Quadratic Family

“…In the presence of R-symmetry, a similar partition holds for σ (see Definitions 2.8, 2.10). Indeed, one can conclude the following from the work of [LLMM18b] together with a variant of Theorem 4.7: Let us discuss cases (1) and (2) in Corollary 1.2. As c ∈ R passes through −.75, the dynamics of p c undergo the well-known phenomenon of bifurcation: the finite attracting fixed point of p c in case (1) "becomes" a period 2 attracting cycle as one passes to case (2).…”

“…The second example (see the left-most curve in Figure 3) is that of a disjoint union of a cardioid with the exterior of a circle. The study of the dynamics of the Schwarz reflection map defined by z → σ(z) was initiated in [LM16], and several works have since studied the dynamics of σ for various classes of quadrature domains: see [LLMM18a], [LLMM18b], [LLMM19a], [LLMM19b], [LMM19], [LMM20]. Associated with σ is a natural dynamical partition of Ĉ.…”

On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections

“…However, in the 1990's Bullet and Penrose [12] discovered a phenomenon of explicit mating of two actions, of a quadratic polynomial with a modular group, induced by a single algebraic correspondence on two parts of its domain. And recently, an abundant supply of similar matings generated by the Schwarz reflection dynamics was produced by Lee and Makarov in collaboration with two of the authors of this paper [27,28,29]. It turns out that this machinery is relevant to the theme of this paper Our main example is the classical Apollonian gasket Λ H , which is the limit set of a Kleinian reflection group H generated by reflections in four pairwise kissing circles, see Figure 1.…”

Bers Slices in Families of Univalent Maps

“…to produce a richer conformal dynamical system in the same class. Examples of "hybrid dynamical systems" that are conformal matings of Kleinian reflection groups and anti-holomorphic rational maps (anti-rational for short) were constructed in [LLMM18a,LLMM18b] as Schwarz reflection maps associated with univalent rational maps. Roughly speaking, this means that the dynamical planes of the Schwarz reflection maps in question can be split into two invariant subsets, on one of which the map behaves like an anti-rational map, and on the other, its grand orbits are equivalent to the grand orbits of a group.…”