Bers slices in families of univalent maps

Abstract

We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions Σ. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial z → z d is the Schwarz reflection map arising from the corresponding map in Σ. We characterize the image of this embedding in Σ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial.

Combining rational maps and Kleinian groups via orbit equivalence

“…Remark In the antiholomorphic world, analogs of Theorem 7.15 were proved in [31] (also compare [28, Theorem B]), where dynamically natural homeomorphisms between limit sets of kissing reflection groups and Julia sets of critically fixed antirational maps were constructed.…”

“…We use instead the theory of David homeomorphisms [20]. The examples in [28, 29, 33] may be viewed as antiholomorphic precursors of the holomorphic mating construction of Theorem A. We should, however, point out that there is no natural antiholomorphic analog of higher Bowen–Series maps for reflection groups.…”

Circle packings, kissing reflection groups and critically fixed anti-rational maps

“…A classification of critically fixed anti-rational maps has also been obtained in [9]. There is also a connection between Kleinian reflection groups, anti-rational maps and Schwarz reflections of quadrature domains explained in [21,24,19,20]. In the holomorphic setting, critically fixed rational maps have been studied in [4,11].…”

Quadrature Domains and the Real Quadratic Family

“…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.…”

“…One readily checks that p c β :ψ(A) → ψ(A) and p : int K(p) → int K(p) are conformally conjugate. Likewise, φ β • φ −1 Γ • ψ −1 : ψ(I) → D \ φ β (U ) defines a conjugacy between p c β : ψ(I) → Ĉ and φ β • ρ • φ −1 β : D \ φ β (U ) → D.For f as in (17), one may mimic the proofs in Section 6 (or Lemma 4.10 of[LMM20]) to show that A and I share a common boundary which is a Jordan curve. Thus the same holds for the common boundary between the filled Julia set and escaping set of p c β.…”