Pith. sign in

REVIEW 3 major objections 3 minor 33 references

The paper proves that pairs of parabolic maps can be mated with free products of cyclic groups via algebraic correspondences that, for the first time, are not conjugate to their own inverses.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 02:17 UTC pith:XDMACOHZ

load-bearing objection A genuine advance in parabolic pair-matings, but Theorem D has an unproved holomorphicity step and the non-time-reversibility claim is asserted without proof. the 3 major comments →

arxiv 2607.14364 v1 pith:XDMACOHZ submitted 2026-07-15 math.DS

Matings between compositions of rational maps and free products of finite cyclic groups

Shaun Bullett , Luna Lomonaco , Miguel Ratis Laude This is my paper
classification math.DS MSC 37F1030F4037F34
keywords matingsalgebraic correspondencesrational mapsfree products of cyclic groupsKleinian groupsquasiconformal surgerypinched polynomial-like mapsdeleted covering correspondences
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs, for any degrees p,q ≥ 2, a holomorphic algebraic correspondence of bidegree (pq,pq) on the Riemann sphere that simultaneously realizes the dynamics of the two compositions g∘f and f∘g and the action of a faithful discrete representation of the free product C_{p+1}*C_{q+1}. The construction works in two settings: parabolic rational maps f,g whose compositions have connected filled Julia set, and arbitrary polynomials with connected filled Julia set mated with Kleinian representations with totally disconnected limit set. In both cases the mating correspondence factors as the composition of two 'deleted covering correspondences' attached to rational maps P,Q that are conjugate to polynomials of degrees p+1 and q+1. The authors' headline claim is that these are the first matings that are not time-reversible: unlike all previously known correspondences of this kind, they are not conjugate to their own inverse.

Core claim

The central discovery is that a mating between a pair of maps (f,g) and the group Γ_{p,q} ≅ C_{p+1}*C_{q+1} can be realized not just by an abstract algebraic correspondence, but concretely as F = Cov^0_Q ∘ Cov^0_P, the composition of the deleted covering correspondences of two rational maps P and Q that are each conjugate to polynomials of degrees p+1 and q+1. This factorization gives the mating an explicit algebraic form and reveals its lack of time-reversibility: reversing F introduces a commutator between the two covering correspondences, rather than landing in the same conjugacy class. The proof glues the pinched polynomial-like restrictions of g∘f and f∘g to the fundamental domains of Γ

What carries the argument

The pinched polynomial-like map: a branched cover from a pinched polygon onto a polygon, used to localize g∘f and f∘g around their filled Julia sets; and the deleted covering correspondence Cov^0_R of a rational map R, defined by R(w)=R(z), w≠z, which sends each point to the other points in its fiber. The group side is the Fuchsian representation Γ_{p,q} of C_{p+1}*C_{q+1}, whose generators have parabolic composition; its fundamental domain is a hyperbolic triangle. The mating is assembled by quasiconformally interpolating boundary homeomorphisms that conjugate the boundary actions of the group to those of the polynomial-like maps, then applying the Measurable Riemann Mapping Theorem.

Load-bearing premise

The proof of Theorem D assumes that the quasiconformal boundary-conjugating maps φ_- and φ_+ become conformal on the tiles where P and Q are defined, so that P and Q are holomorphic; quasiconformal conjugates of Möbius transformations are not generally holomorphic, and no conformality is established.

What would settle it

Compute the correspondence F for a specific example (e.g., f(z)=z^2/(z+1/2), g(z)=(2z^2-z-1/2)/(2z+1) from Figure 1) and compare the iterated images of a generic point under F and under F^{-1}; if the closures are the same, time-reversibility would hold, contradicting the paper's claim. Alternatively, verify directly on the tiles φ_-(ρ^{-m}(A_ε∩Δ_ρ)) whether φ_- is conformal; a single tile where the quasiconformal dilatation is nonzero disproves the holomorphicity step in the proof of Theorem D.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every pair f,g satisfying the hypotheses yields an explicit algebraic correspondence F of bidegree (pq,pq), given in closed form as Cov^0_Q ∘ Cov^0_P.
  • The filled Julia sets K(g∘f) and K(f∘g) are glued together in the mating, with only the parabolic fixed point identified; the correspondence F|Λ- is hybrid conjugate to g∘f and F^{-1}|Λ+ to f∘g.
  • Because F is not conjugate to F^{-1}, the standard symmetry argument used to classify previous matings does not apply; new non-reversible dynamical systems are obtained.
  • For polynomials, the construction extends to all faithful discrete representations of C_{p+1}*C_{q+1} with Cantor limit set, via a Kleinian perturbation of Γ_{p,q} in which all non-elliptic elements are hyperbolic.
  • The factorization theorem shows that the mating correspondence is determined by the two polynomial-conjugate maps P,Q, so the parameter space of matings is a quotient of the product of the degree-(p+1) and degree-(q+1) polynomial spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the factorization is correct, the non-reversibility should be visible numerically: the correspondence F and its inverse have different grand orbit structures, so iterating F versus F^{-1} should produce different limit sets for generic starting points; a computational check on the examples in Figure 1 would test this directly.
  • The construction hints at a broader principle: any pair of maps with a common 'pinching' structure and a group with matching orbifold might admit a mating correspondence of the form Cov^0_Q ∘ Cov^0_P, suggesting a systematic recipe for building matings between iterated compositions and free products.
  • A direct test of the proof's load-bearing step would be to check whether φ_- is conformal on the tiles φ_-(ρ^{-m}(A_ε∩Δ_ρ)); if not, the non-parabolic orbit matings of Theorem D may still exist but require a different factorization argument.
  • The examples with f=g (Figure 8, left) show that even self-mated pairs can yield non-time-reversible correspondences, implying the phenomenon is not merely an artifact of asymmetry between f and g.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper constructs algebraic correspondences on the Riemann sphere that mate a pair of rational maps (f,g), each with a parabolic fixed point, with the Fuchsian group Γ_{p,q} isomorphic to C_{p+1}*C_{q+1}. The main parabolic result (Theorem A) produces a holomorphic correspondence of bidegree (pq,pq) under the hypothesis that g∘f belongs to Per^{pq}_1(1) and has connected filled Julia set. Theorem B represents such a correspondence as the composition of two deleted covering correspondences of rational maps conjugate to polynomials. Theorems C and D extend these constructions to polynomial pairs and to non-parabolic faithful discrete Kleinian representations with totally disconnected limit set, using the weaker notion of "orbit mating." The paper also claims that these are the first matings that are not time-reversible.

Significance. If the main theorems are correct, this is a substantial extension of mating theory: instead of mating a single map with a group, the paper mates a pair of maps with a free product of finite cyclic groups. The construction is explicit and uses a combination of pinched polynomial-like restrictions, group-combinatorial gluing, and quasiconformal/Beltrami surgery. The parabolic part (Theorems A and B) is detailed and draws on established techniques, and the algebraic correspondence framework is appropriate. The claimed factorizations into deleted covering correspondences are of independent interest, and the potential non-reversibility of the new correspondences would be a notable novelty. However, the proof of Theorem D contains a load-bearing gap, and a headline claim in the abstract is not proved.

major comments (3)
  1. [§4.3, Proof of Theorem D] The factorization F = Cov_Q^0 ∘ Cov_P^0 is not established. The map P is declared to be holomorphic on the tiles φ_-(ρ^{-m}(A_ε∩Δ_ρ)) because "φ_- conjugates the action of ρ_ε to a holomorphic map on Ĉ." This does not follow. Lemma 4.2 supplies only quasiconformal extensions φ_-, φ_+ of boundary diffeomorphisms, and a quasiconformal conjugate of a Möbius transformation need not be holomorphic. The Beltrami straightening in Theorem C makes the global correspondence F holomorphic, but it does not make φ_- or φ_+ conformal on the tiles; their Beltrami coefficients are not shown to vanish. In Theorem B the analogous construction worked because the definition of mating gives a conformal conjugacy on Ω, whereas in Theorem C/D only a grand-orbit equivalence is available. Thus P and Q are not proved to be rational, and Theorem D's central claim is unsupported. The authors need either to prove th
  2. [Theorems B and D, statements] Both theorem statements say that P and Q have degrees p and q, respectively. The proofs and the abstract require degrees p+1 and q+1. As stated, Cov_P^0 and Cov_Q^0 would have bidegrees (p-1,p-1) and (q-1,q-1), so Cov_Q^0∘Cov_P^0 would have bidegree ((p-1)(q-1),(p-1)(q-1)), not the asserted (pq,pq). This appears to be a typo, since the proof constructs P of degree p+1 and Q of degree q+1, but the statements must be corrected.
  3. [Abstract and Introduction] The paper advertises that these matings are "the first examples that are not time-reversible." No proof of non-reversibility is supplied. Showing that the construction is not of the form J∘Cov_R^0 does not imply that the correspondence is not conjugate to its own inverse. Either prove non-reversibility for at least the constructed family, or qualify the claim as "not of the previously known reversible form."
minor comments (3)
  1. [Theorem D statement] The group is written as C_p*C_q; it should be C_{p+1}*C_{q+1} to match Theorem C, the abstract, and the rest of the paper.
  2. [Lemma 3.2] The proof of the quasiconformal extension lemma is condensed and refers to [9] and [14] for the main method. Since Theorem A depends on it, please expand the cusp-extension argument or give precise references to the exact statements being used.
  3. [Proof of Theorem B] In the definition of P on Λ_-, the expression ϕ_+^{-1}∘ f∘ϕ_- is not explained; a sentence clarifying that f sends K(g∘f) to K(f∘g), so the composition lands in Λ_+, would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the main construction is self-contained, and the self-citations are background rather than load-bearing.

full rationale

The results are constructive existence theorems. Theorem A assembles a topological correspondence G from pinched polynomial-like restrictions of g∘f and f∘g and from the explicit action of Γ_{p,q}, then straightens G with an invariant Beltrami form. Theorem C does the same with polynomial-like restrictions and an annulus quotient. The factorization results (Theorems B and D) are not predictions from fitted data: the maps P and Q are explicitly defined from the conjugacy maps φ, φ± and from f,g, and the equality Cov_Q^0 ∘ Cov_P^0 = F is verified on an open set with an accumulation point and then extended analytically. This is construction, not a fit renamed as a prediction. Self-citations appear — [9] for the pinched polynomial-like/Farey-like framework, [14] for the method in Lemma 3.2, [5,30] for the orbit-mating notion — but they provide definitions or parallel techniques; the load-bearing quasiconformal extension in Lemma 3.2 is argued directly using Pommerenke and Warschawski estimates. The genuine local weakness is in §4.3, where the paper asserts that P is holomorphic on tiles since φ_- conjugates the action of ρ_ε to a holomorphic map on Ĉ, although φ_- is only quasiconformal. This is a correctness gap in the proof of Theorem D, but it is not a circularity: it does not reduce Theorem D to its own assumptions. Hence the circularity score is minimal.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

Pure mathematics paper: no empirical data. The construction uses standard tools (Measurable Riemann Mapping, Chow, Klein combination) and a few results from earlier papers by the same group. No free parameters are fitted: the angle θ, neighborhoods U,V, and perturbation ε are auxiliary construction choices, not fitted values. The main unproved-but-needed input is the quasiconformal rigidity of the Kleinian representations and the technical gluing lemma deferred to [9].

axioms (5)
  • standard math Measurable Riemann Mapping Theorem
    Used in proofs of Theorems A and C to straighten the topological correspondence G into a holomorphic correspondence F.
  • standard math Chow's theorem: closed analytic subsets of complex projective space are algebraic
    Used in §2.3 and Theorem A to conclude that the straightened holomorphic correspondence is an algebraic correspondence.
  • standard math Klein combination theorem for free products of finite cyclic groups
    Used in §2.2 to show that Γ_{p,q} is a faithful discrete representation of C_{p+1}*C_{q+1}.
  • standard math Quasiconformal extension and Warschawski strip estimates
    Used in Lemma 3.2 to extend boundary diffeomorphisms quasiconformally across cusps; the proof is condensed and partly deferred to [9, Lemma 5.2].
  • domain assumption Quasiconformal rigidity of faithful discrete representations of C_{p+1}*C_{q+1} with Cantor limit set
    Invoked in §4.3 to reduce to the Fuchsian representative Γ_ε; a known theorem in Kleinian groups, not proved in this paper.

pith-pipeline@v1.3.0-alltime-deepseek · 20439 in / 18356 out tokens · 159484 ms · 2026-08-02T02:17:33.370677+00:00 · methodology

0 comments
read the original abstract

Given a pair of rational maps $(f, g)$, of degrees $p$ and $q$, each with a parabolic fixed point having a fully invariant simply-connected basin of attraction, we construct an algebraic correspondence $F$ on the Riemann sphere, of bidegree $(pq, pq)$, realizing a mating between the two compositions $g\circ f$ and $f\circ g$ of the maps, and the parabolic faithful discrete representation of the free product of cyclic groups of orders $p + 1$ and $q + 1$. We also show that $F$ is the composition of a pair of deleted covering correspondences of rational maps which are conjugated to polynomials of degrees $p + 1$ and $q + 1$. We generalize our method to construct matings between compositions of pairs of polynomials and (non-parabolic) faithful Kleinian representations of the same group, now with connected regular set. As far as we are aware, these matings between pairs of maps and groups are the first examples that are not time-reversible (that is, they are not conjugate to their own inverses).

Figures

Figures reproduced from arXiv: 2607.14364 by Luna Lomonaco, Miguel Ratis Laude, Shaun Bullett.

Figure 1
Figure 1. Figure 1: The mating between the pair (f, g) of quadratic maps f(z) = z 2/(z + 1/2) and g(z) = (2z 2 − z − 1/2)/(2z + 1), and the group Γ2,2, isomorphic to C3 ∗ C3. hyperelliptic surfaces as well as on the Riemann sphere. Our first results concern the parabolic side of the story. We will introduce the parabolic families Pd and Perd 1 (1), and the group Γp,q, which is a representation of Cp+1∗Cq+1 in P SL(2, R) such … view at source ↗
Figure 2
Figure 2. Figure 2: The pinched polynomial-like restrictions of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On the left, the fundamental domain for ρ, and on the right, the fundamental domain for σ. In the middle, the intersection between these domains, a fundamental domain for Γp,q = ⟨ρ, σ⟩, together with some of its images under elements of the group. for the action of ρ on D, and similarly the domain ∆σ ⊂ D of points to the left of the geodesic rays starting at Q and ending at ±i is a fundamental domain for t… view at source ↗
Figure 4
Figure 4. Figure 4: The geodesic I and its images under different elements of Γp,q. The curves in blue and red are left from the tessellation of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The adapted picture of the group Γp,q. iterates of ρ, like the p : 1 map f : ∂U′ → ∂Vˆ ; a q : 1 map from ∂iBˆ to γ defined by the iterates of σ, like the q : 1 map g : ∂Vˆ → ∂U; a q : 1 map from ∂iB to ∂iAˆ defined by the iterates of σ, like the q : 1 map g : ∂V ′ → ∂Uˆ; and a p : 1 map from ∂iAˆ to η defined by the iterates of ρ, like the p : 1 map f : ∂Uˆ → ∂V . See [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 6
Figure 6. Figure 6: The Farey-like map E1 : X′ → X induced from the action of the elements σ mρ n on the left side of the unit disk. The gray regions show the sets A and X \ X′ . Notice that the angle outside of X is 2θ. Proof. Let us define the half-disks Dℓ := {x + iy ∈ D | x ≤ 0} and Dr := {x + iy ∈ D | x ≥ 0}; notice that A ⊂ Dℓ and B ⊂ Dr. The map z 7→ −z 2 maps both half-disks to the whole of D, sending ±i to 1. Let X b… view at source ↗
Figure 7
Figure 7. Figure 7: The maps presented in the proof of Lemma 3.2. The key for this result is the control [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Examples of matings with the group Γ2,2. On the right is the same example as [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Constructions in the proof of Theorem B. The tile [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The polynomial-like restrictions of g ◦ f and f ◦ g, and the intermediary domains induced by f and g. a larger simply connected neighborhood of K(f ◦ g) such that, setting V ′ := (f ◦ g) −1 (V ), one has V ′ ⋐ f(U ′ ) ⋐ V . See [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The fundamental domain ∆ε for the action of the perturbed group Γ ε p,q. The first images of the boundary of ∆ε by elements of the group are also depicted. We return now to the action of Γε on the Poincaré 2-disk D. As in the case of Γp,q, we denote the geodesic connecting i to −i by I, and, as in that case, we have I ⊂ ∆ε, but now the images of I under the group elements σ n ε ρ m ε (respectively ρ m ε σ… view at source ↗
Figure 12
Figure 12. Figure 12: An example of the identifications that convert each set [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The annuli A and B, and the induced actions of ρ and σ on their boundaries and intermediary curves. The precise power of ρ or σ varies along these curves. Notice the similarity with [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 4 linked inside Pith

  1. [1]

    A. F. Beardon. Polynomials with identical Julia sets.Complex Variables, Theory and Application: An International Journal, 17(3-4):195–200, 1992

    1992

  2. [2]

    A.F. Beardon. Symmetries of Julia sets.Bulletin of the London Mathematical Society, 22(6):576–582, 1990

    1990

  3. [3]

    S. R. Bullett and M. Freiberger. Holomorphic correspondences mating Chebyshev-like maps with Hecke groups.Ergodic Theory and Dynamical Systems, 25(4):1057–1090, 2005

    2005

  4. [4]

    S. R. Bullett and P. Haïssinsky. Pinching holomorphic correspondences.Conformal Geom- etry and Dynamics of the American Mathematical Society, 11:65–89, 2007

    2007

  5. [5]

    S. R. Bullett and W.J. Harvey. Mating quadratic maps with Kleinian groups via quasicon- formal surgery.Electronic Research Announcements of the American Mathematical Society, 6(3):21–30, 2000

    2000

  6. [6]

    S. R. Bullett and L. Lomonaco. Mating quadratic maps with the modular group II.Inven- tiones mathematicae, 220:185–210, 2019

    2019

  7. [7]

    S. R. Bullett and L. Lomonaco. Dynamics of modular matings.Advances in Mathematics, 410, 2022

    2022

  8. [8]

    S. R. Bullett and L. Lomonaco. Mating quadratic maps with the modular group III: The modular mandelbrot set.Advances in Mathematics, 458, 2024

    2024

  9. [9]

    S. R. Bullett, L. Lomonaco, M. Lyubich, and S. Mukherjee. Mating parabolic rational maps with Hecke groups.https://arxiv.org/abs/2407.14780

    arXiv

  10. [10]

    S. R. Bullett and C. Penrose. Mating quadratic maps with the modular group.Inventiones mathematicae, 115:483–512, 1994. 25

    1994

  11. [11]

    S. R. Bullett and C. Penrose. Regular and limit sets for holomorphic correspondences. Fundamenta Mathematicae, 167:111–171, 2001

    2001

  12. [12]

    W.-L. Chow. On Compact Complex Analytic Varieties.American Journal of Mathematics, 71(4):893–914, 1949

    1949

  13. [13]

    Douady and J.H

    A. Douady and J.H. Hubbard. On the dynamics of polynomial-like mappings.Annales scientifiques de l’École Normale Supérieure, 18(4):287–343, 1985

    1985

  14. [14]

    S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. Schwarz reflections and anti- holomorphic correspondences.Advances in Mathematics, 385:107766, 2021

    2021

  15. [15]

    S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. Dynamics of Schwarz reflections: the mating phenomena.Annales Scientifiques de l’École Normale Supérieure, 56(6):1826– 1881, 2023

    2023

  16. [16]

    S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. Schwarz reflections and the tricorn.Annales de l’Institut Fourier, 75(5):1987–2100, 2025

    1987

  17. [17]

    Levin and F

    G. Levin and F. Przytycki. When do two rational functions have the same Julia set? Proceedings of the American Mathematical Society, 154(6):2179–2190, 1997

    1997

  18. [18]

    Lomonaco

    L. Lomonaco. Parameter space for families of parabolic-like mappings.Advances in Math- ematics, 261:200–219, 2014

    2014

  19. [19]

    Lomonaco

    L. Lomonaco. Parabolic-like mappings.Ergodic Theory and Dynamical Systems, 35(7):2171–2197, 2015

    2015

  20. [20]

    Y. Luo, M. Lyubich, and S. Mukherjee. A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences.https://arxiv.org/abs/2408.00204

    Pith/arXiv arXiv

  21. [21]

    Lyubich, J

    M. Lyubich, J. Mazor, and S. Mukherjee. Antiholomorphic correspondences and mating II: Shabat polynomial slices.https://arxiv.org/abs/2509.12357

    arXiv

  22. [22]

    Lyubich, J

    M. Lyubich, J. Mazor, and S. Mukherjee. Antiholomorphic correspondences and mating I: Realization theorems.Communications of the American Mathematical Society, 4:495–547, 2024

    2024

  23. [23]

    Lyubich, S

    M. Lyubich, S. Merenkov, S. Mukherjee, and D. Ntalampekos. David extension of circle homeomorphisms, welding, mating, and removability.Memoirs of the American Mathemat- ical Society, 313(1588):v+110, 2025

    2025

  24. [24]

    Lyubich and S

    M. Lyubich and S. Mukherjee. Mirrors of conformal dynamics: Interplay between anti- rational maps, reflection groups, Schwarz reflections, and correspondences.https://arxiv. org/abs/2310.03316

    Pith/arXiv arXiv

  25. [25]

    Pinchingandplumbingdeformationsofquadraticrationalmaps

    P.Makienko. Pinchingandplumbingdeformationsofquadraticrationalmaps. International Centre for Theoretical Physics, Trieste (Italy),https://inis.iaea.org/collection/ NCLCollectionStore/_Public/24/036/24036519.pdf, 1993. 26

    1993

  26. [26]

    B. Maskit. On Klein’s combination theorem.Transactions of the American Mathematical Society, 120(3):499–509, 1965

    1965

  27. [27]

    Mj and S

    M. Mj and S. Mukherjee. Matings, holomorphic correspondences, and a Bers slice.Journal de l’École polytechnique — Mathématiques, 12:1445–1502, 2025

    2025

  28. [28]

    Mukherjee and S

    S. Mukherjee and S. Viswanathan. Correspondences on hyperelliptic surfaces, combination theorems, and Hurwitz spaces.https://arxiv.org/abs/2508.18711

    Pith/arXiv arXiv

  29. [29]

    Pommerenke.Boundary Behaviour of Conformal Maps

    C. Pommerenke.Boundary Behaviour of Conformal Maps. Springer Berlin, Heidelberg, 1992

    1992

  30. [30]

    Ratis Laude

    M. Ratis Laude. Continuity of matings of Kleinian groups and polynomials.https:// arxiv.org/abs/2411.08748

    Pith/arXiv arXiv

  31. [31]

    J.F. Ritt. Periodic Functions with a Multiplication Theorem.Transactions of the American Mathematical Society, 23(1):16–25, 1922

    1922

  32. [32]

    J.F. Ritt. Permutable Rational Functions.Transactions of the American Mathematical Society, 25(3):399–448, 1923

    1923

  33. [33]

    S. E. Warschawski. On conformal mapping of infinite strips.Transactions of the American Mathematical Society, 51(2):280–335, 1942. 27

    1942