Tessellating the discreteness locus for the modular mating family of correspondences

Abstract: The modular Mandelbrot set $M_Γ$, the connectedness locus of the modular mating family of 2 : 2 holomorphic correspondences $\mathcal{F}<em>a$ on the Riemann sphere, is homeomorphic to the classical Mandelbrot set $M$. The Klein combination locus $\mathcal{K}$ (the "discreteness locus" of the family $\mathcal{F}_a$) is a pinched neighborhood of $M</em>Γ$ in the $a$-plane, pinched at the root point. We construct a canonical map $Ψ$ from $\mathcal{K}\setminus M_Γ$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $$Φ: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$$, and we prove that $Ψ$ is analytic. This map $Ψ$ induces a tessellation of $\mathcal{K}\setminus M_Γ$ by pulling back a tessellation of $\mathbb{H}$ invariant under the modular group. We develop a series of conjectures concerning the structure of $\mathcal{K}$, its boundary, and $Ψ(\mathcal{K}) \subset \mathbb{H}$.

• The paper constructs a canonical map Ψ from a pinched neighbourhood of the modular Mandelbrot set into the hyperbolic plane and proves its branches are locally analytic using holomorphic motion and modular-surface rigidity.
• Pulling back modular tessellations through Ψ creates a Douady–Hubbard-style combinatorial framework, while critical-relation parameters in the Klein combination locus correspond precisely to torsion points of the modular surface.
• The paper establishes the first pinching and boundary cases, identifies monodromy around the puncture at a=1, and formulates conjectures about the global topology of the discreteness locus and its boundary parameters.

This paper studies the parameter space of the family $\mathcal{F}_a$ of $(2{:}2)$ holomorphic correspondences on $\widehat{\mathbb{C}}$ introduced by Bullett and Penrose, whose connectedness locus — the modular Mandelbrot set $M_\Gamma$ — is homeomorphic to the classical Mandelbrot set. The authors construct a canonical dynamically defined map $\Psi$ from a pinched neighbourhood of $M_\Gamma$ into the hyperbolic plane, prove its analyticity, and use it to pull back modular tessellations of $H$, thereby producing an analogue of the Douady–Hubbard external ray/equipotential framework for this correspondence family (2608.17243).

Background: the family $\mathcal{F}_a$ and the Klein combination locus

In suitable coordinates the correspondence is $\mathcal{F}_a = J_a \circ Cov_0^Q$, where $J_a$ is the Möbius involution with fixed points $(2{:}2)$0 and $(2{:}2)$1, and $(2{:}2)$2 is the deleted covering correspondence of the cubic $(2{:}2)$3. The point $(2{:}2)$4 is a parabolic fixed point of a branch of $(2{:}2)$5. The Klein combination condition requires fundamental domains $(2{:}2)$6 for $(2{:}2)$7 and $(2{:}2)$8 for $(2{:}2)$9 whose union is $\widehat{\mathbb{C}}$0; imposing transversality of their common tangent at $\widehat{\mathbb{C}}$1 against the attracting–repelling axis yields a well-defined partition into regular set $\widehat{\mathbb{C}}$2 and limit set $\widehat{\mathbb{C}}$3. The Klein combination locus $\widehat{\mathbb{C}}$4 consists of parameters admitting such a pair; it plays the role of a discreteness locus, analogous to the discreteness locus for representations of $\widehat{\mathbb{C}}$5 in $\widehat{\mathbb{C}}$6. For $\widehat{\mathbb{C}}$7, $\widehat{\mathbb{C}}$8 is a mating of a parabolic quadratic rational map with the modular group $\widehat{\mathbb{C}}$9, and the grand orbit space $M_\Gamma$0 is conformally a thrice-marked sphere, hence canonically isomorphic to the modular surface $M_\Gamma$1.

Outside $M_\Gamma$2, the dynamics appears generically chaotic: no regular/limit set partition exists, hence no Böttcher-type conjugacy. The paper notes (without proof) that for a dense set of parameters in $M_\Gamma$3 the grand orbit space should have no Hausdorff component.

Construction of the canonical map $M_\Gamma$4

The central observation parallels Douady and Hubbard's proof that the Mandelbrot set is connected. For $M_\Gamma$5 with the standard Klein combination pair, the partial Böttcher map $M_\Gamma$6 is built on $M_\Gamma$7 by lifting paths from the fixed point of $M_\Gamma$8 to paths in $M_\Gamma$9 starting at the fixed point of $\Psi$0, then extended equivariantly to tiles containing the critical value $\Psi$1. Setting $\Psi$2 yields a canonical continuous map $\Psi$3: canonicity means that $\Psi$4 maps to $\Psi$5 under the unique marked conformal isomorphism $\Psi$6, so $\Psi$7 is determined up to the discrete fibre $\Psi$8.

On the real interval $\Psi$9 the image is explicit: $M_\Gamma$0, $M_\Gamma$1, with limits $M_\Gamma$2 as $M_\Gamma$3 (the Misiurewicz point) and $M_\Gamma$4 as $M_\Gamma$5. Pulling back any $M_\Gamma$6-invariant tessellation of $M_\Gamma$7 through $M_\Gamma$8 produces a combinatorial skeleton of $M_\Gamma$9 analogous to the Douady–Hubbard grid; computer plots show tile vertices coincide across different choices of fundamental domain, though tile edges cannot yet be plotted exactly because the canonical marking lines on the orbifold are not algorithmically accessible.

Extension past the puncture and multivaluedness

The domain extends beyond $H$0 to discs $H$1 via explicit families of Klein combination pairs, and to the disc $H$2 by homotoping the standard pair around the puncture at $H$3 (where $H$4 is undefined). A corollary is that $H$5: the point $H$6 is a critical point of $H$7 and so cannot lie in the interior of any transversal.

The puncture makes extensions multivalued. Extending along upper versus lower paths around $H$8 gives branches $H$9 and $\mathcal{F}_a$0 related by $\mathcal{F}_a$1; both converge to $\mathcal{F}_a$2-orbit points as $\mathcal{F}_a$3, namely $\mathcal{F}_a$4 and $\mathcal{F}_a$5 respectively. This monodromy is a structural difference from the Douady–Hubbard setting, where $\mathcal{F}_a$6 is single-valued on the simply connected complement of $\mathcal{F}_a$7.

At $\mathcal{F}_a$8 itself, although $\mathcal{F}_a$9, the correspondence remains discrete: the sphere partitions into a tiled open set $\mathcal{F}_a = J_a \circ Cov_0^Q$0 and a Cantor limit set contained in $\mathcal{F}_a = J_a \circ Cov_0^Q$1. This "pinching" mechanism — contracting lifts of a path on the orbifold — generalizes to the boundary parameters discussed below.

Analyticity

The main theorem states that every branch of every canonical map $\mathcal{F}_a = J_a \circ Cov_0^Q$2 is locally analytic on its domain in $\mathcal{F}_a = J_a \circ Cov_0^Q$3. The proof circumvents the absence of a superattracting organizing centre (infinity, in the polynomial case) by exploiting conformal rigidity of the modular surface: a holomorphic motion of the fundamental domains (constructed via pre-Fatou coordinates at the parabolic point and Slodkowski extension), pushed through the quotient maps, produces an analytic family of $\mathcal{F}_a = J_a \circ Cov_0^Q$4-invariant Beltrami coefficients on $\mathcal{F}_a = J_a \circ Cov_0^Q$5; integrating them yields quasiconformal maps normalizing $\mathcal{F}_a = J_a \circ Cov_0^Q$6 to itself, which must be conformal since $\mathcal{F}_a = J_a \circ Cov_0^Q$7 is quasiconformally rigid. Discreteness of the critical-relation set $\mathcal{F}_a = J_a \circ Cov_0^Q$8 plus Riemann removable singularities handles the remaining points.

A second consequence of the holomorphic motion machinery is a rigidity statement: within each connected component of $\mathcal{F}_a = J_a \circ Cov_0^Q$9, all correspondences $J_a$0 are quasiconformally conjugate, while points of $J_a$1 together with centers of hyperbolic components are quasiconformally rigid. This mirrors the classical picture of the Mandelbrot set's internal structure, transferred to the discreteness locus.

Critical relations

A critical relation parameter is one where the grand orbit of the critical point $J_a$2 contains the fixed point $J_a$3 of $J_a$4 (a 2-relation) or $J_a$5 of $J_a$6 (a 3-relation); using reversibility ("maps of triples"), every such relation reduces to forwards or backwards form. Inside $J_a$7, only forwards relations occur, and they correspond exactly to torsion points of $J_a$8: $J_a$9 is stabilized by a non-trivial elliptic subgroup if and only if $(2{:}2)$00 satisfies a critical relation. Computer plots suggest all critical relations inside $(2{:}2)$01 lie within $(2{:}2)$02.

Outside $(2{:}2)$03, backwards relations abound; the plot of such parameters suggests they accumulate everywhere on $(2{:}2)$04 and possibly throughout the complement. Each backwards relation of length $(2{:}2)$05 completes to a parabolic cycle of length $(2{:}2)$06 (2-relations) or $(2{:}2)$07 (3-relations), passing through the critical point but not the critical value.

Conjectured structure of $(2{:}2)$08 and $(2{:}2)$09

The paper formulates two principal conjectures. First, for each $(2{:}2)$10 there is a unique backwards 2-relation parameter $(2{:}2)$11 (with conjugates $(2{:}2)$12), satisfying: $(2{:}2)$13; pinched fundamental domains meeting only at the parabolic cycle and $(2{:}2)$14; discreteness of $(2{:}2)$15; and convergence $(2{:}2)$16. The case $(2{:}2)$17 ($(2{:}2)$18) is proved here, and $(2{:}2)$19 was established previously by Bullett and Curtis; general $(2{:}2)$20 would follow if the inclusion $(2{:}2)$21 into the set of accessible points were surjective. Whether any backwards 3-relations lie on $(2{:}2)$22 is open; the candidate $(2{:}2)$23 provably admits no suitable fundamental-domain pair.

Second, the global structure conjecture asserts that $(2{:}2)$24 is connected (a topological punctured disc with inward cusps at $(2{:}2)$25 and $(2{:}2)$26), that $(2{:}2)$27 is simply connected after cutting along $(2{:}2)$28, and that $(2{:}2)$29 extends uniquely and injectively to this cut domain. A schematic scenario identifies $(2{:}2)$30 with a curve in $(2{:}2)$31 mapping $(2{:}2)$32 to an interval of $(2{:}2)$33, with torsion points $(2{:}2)$34 accumulating at the root. The analogy with the $(2{:}2)$35 discreteness locus is close but imperfect: pleating rays fill the group-theoretic locus for all rational rotation numbers, whereas here only rotation numbers $(2{:}2)$36 give conjectural rays — a discrete rather than dense family.

Limitations and open questions

Several results rest on unproved assumptions or remain partial. The density claims for critical relation parameters outside $(2{:}2)$37, and the assertion that generic chaotic parameters have no Hausdorff orbit-space component, are supported only by numerics. The conjectured convergence of $(2{:}2)$38 near arbitrary boundary points, the existence and uniqueness of the $(2{:}2)$39, and the global topology of $(2{:}2)$40 are all conjectural. No algorithm exists for plotting true pull-back tessellation edges, limiting numerical verification. Three specific questions are posed: whether $(2{:}2)$41 has the local form $(2{:}2)$42 with $(2{:}2)$43 real-analytic near the puncture; the structure of $(2{:}2)$44 between consecutive $(2{:}2)$45; and whether isolated discrete correspondences outside $(2{:}2)$46 (such as the harmonic sequence at parameters $(2{:}2)$47 where neutral fixed points rotate by $(2{:}2)$48, proved discrete for $(2{:}2)$49 by Curtis) are isolated or belong to stable regions.

Conclusion

The paper transfers the Douady–Hubbard paradigm to the modular mating family of correspondences: a dynamically defined map $(2{:}2)$50, proved analytic via rigidity of the modular surface, tessellates the complement of $(2{:}2)$51 within the Klein combination locus and organizes its critical relations as torsion points of $(2{:}2)$52. The price of losing the superattracting fixed point is a smaller domain of definition — the pinched discreteness locus $(2{:}2)$53 — and monodromy around the puncture at $(2{:}2)$54. The resulting framework reduces the geometry of $(2{:}2)$55 to concrete questions about parabolic cycles and based fundamental domains for $(2{:}2)$56, several of which the paper resolves in low cases and formulates precisely in general.