Pith. sign in

REVIEW 5 cited by

A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences

Not yet reviewed by Pith; the record is open.

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

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2408.00204 v1 pith:HQQTTFRJ submitted 2024-08-01 math.DS math.CVmath.GR

Yusheng Luo , Mikhail Lyubich , Sabyasachi Mukherjee This is my paper · ORCID
classification math.DSmath.CVmath.GR
keywords anti-matingsmapspolynomialsalgebraiccorrespondencesgroupsparameter
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
read the original abstract

In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. These matings emerge naturally as degenerate (anti-)polynomial-like maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps. Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou's vision of a unified theory of groups and maps.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Transcendental correspondences: when Fuchsian groups take over basins of entire maps

    math.DS 2026-07 accept novelty 8.0 of 10

    The authors construct (∞:∞) holomorphic correspondences mating transcendental entire maps with Fuchsian groups, realized as deleted covering correspondences of meromorphic functions with one simple pole.

  2. Uniformization of gasket Julia sets

    math.DS 2024-11 conditional novelty 8.0 of 10

    A gasket Julia set is quasiconformally uniformizable by a round gasket exactly when its Fatou components meet tangentially, and David-uniformizable exactly when all Fatou components are quasidisks.

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

    math.DS 2026-07 conditional novelty 7.0 of 10

    Correspondences are constructed that mate pairs of rational maps with free products of cyclic groups, and factor as compositions of two deleted covering correspondences.

  4. Combining cusped triangle groups with Blaschke products: commensurable matings

    math.DS 2026-07 accept novelty 6.5 of 10

    Algebraic correspondences exist that combine Fuchsian (p,q,∞)-triangle groups with Blaschke products B1=β2,1∘β1,2 and B2=β1,2∘β2,1 of degrees (p-1)(q-1) fixing 0 and 1.

  5. Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

    math.DS 2025-11 unverdicted novelty 1.0 of 10

    Algebraic correspondences mate rational maps with Kleinian groups, and the modular Mandelbrot set is homeomorphic to the Mandelbrot set—this survey reports those results.

Pith tools