arXiv subjects

Sabyasachi Mukherjee

Publications and source records attributed to Sabyasachi Mukherjee.

At least 19 recordsLinked to original sources

arXiv2026

Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets

We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form: \begin{align*} f_{a,b}(x)=\left(2x+a+\dfrac{b}π \sin(2πx)\right) \mod 1,\ a \in \mathbb{R}/\mathbb{Z},\ 0 \leq b \leq 1. \end{align*} We prove simple connectedness of tongues by providing a dynamically natural real-analytic uniformization for each tongue. For maps in a tongue, we characterize the unique maximal subset of the circle on which $f_{a,b}$ is Devaney chaotic. We also show that the Hausdorff dimension of this maximal chaotic set varies real-analytically inside a tongue.

math.DS↗

arXiv2026

Combining cusped triangle groups with Blaschke products: commensurable matings

In this note, we construct algebraic correspondences as matings of Fuchsian $(p,q,\infty)$-triangle groups with Blaschke products. Combined with the results of [MM25], this proves mateability of all cusped triangle groups with suitable Blaschke products. The proof of the main result involves associating two piecewise analytic circle maps to the $(p,q,\infty)-$triangle group, mating these maps with appropriate Blaschke products to produce two commensurable conformal matings, and finally constructing the desired algebraic correspondence as a common lift of the two conformal matings.

math.DS↗

arXiv2026

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

In this paper, we initiate a systematic study of $(\infty : \infty)$ holomorphic correspondences that naturally arise as conformal combinations (matings) of transcendental entire maps with Fuchsian groups. This construction parallels the recent theory of finite-degree algebraic correspondences associated with rational maps. Our correspondence combines the dynamics of a transcendental entire function outside a distinguished attracting/parabolic basin with the action of a compatible Fuchsian group within it. We show that the resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function having exactly one simple pole. When the transcendental entire function has finitely many singular values, so does this meromorphic function, and its line complex can be described explicitly.

math.DS↗

arXiv2026

Mirrors of conformal dynamics: Interplay between anti-rational maps, reflection groups, Schwarz reflections, and correspondences

The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic correspondences obtained by matings between anti-rational maps and reflection groups, and examples of Julia set realizations for limit sets of reflection groups (including classical Apollonian-like gaskets). We follow up these examples with dynamical relations between explicit Schwarz reflection parameter spaces and parameter spaces of anti-rational maps and of reflection groups. These are complemented by a number of general results and illustrations of important technical tools, such as David surgery and straightening techniques. We also collect several analytic applications of the above theory.

math.DS↗

arXiv2026

From Flow to Form: Emergence of the Cytokinetic Ring via Active Cortical Dynamics

During cell division active flows occur in the cortex, a thin layer of gel like network of acto myosin filaments, beneath the cell surface. The cortical flow and the associated stresses bring about change in the cell shape, in particular a sharp invagination at the mid cell. Using 3D phase field simulation of an active deformable shell, which captures coupled dynamics of cortical velocity and nematic order, we show how a nematic like actomyosin ring spontaneously emerge at the equator and drive sharp invagination. We further demonstrate how different cortical flow patterns, including counter rotating flows emerge near the division furrow. We show that these flow patterns, often attributed to intrinsic chirality of actomyosin filaments can instead arise from bias in the initial nematic alignment, revealing a memory effect in the system. By analyzing a simpler model of activity gradient driven compressive flow on a flat interface we decipher the main ingredients for surface instability leading to invagination and counter moving flows.

cond-mat.soft↗

arXiv2026

Teichmüller spaces, polynomial loci, and degeneration in spaces of algebraic correspondences

We develop an analog of the notion of a character variety in the context of algebraic correspondences. It turns out that matings of certain Fuchsian groups and polynomials are contained in this ambient character variety. This gives rise to two different analogs of the Bers slice by fixing either the polynomial or the Fuchsian group. The Bers-like slices are homeomorphic copies of Teichmüller spaces or combinatorial copies of polynomial connectedness loci. We show that these slices are bounded in the character variety, thus proving the analog of a theorem of Bers. To produce compactifications of the Bers-like slices, we initiate a study of degeneration of algebraic correspondences on trees of Riemann spheres, revealing a new degeneration phenomenon in conformal dynamics. There is no available analog of Sullivan's 'no invariant line field' theorem in our context. Nevertheless, for the four times punctured sphere, we show that the compactifications of Teichmüller spaces are naturally homeomorphic.

math.DS↗

arXiv2026

Mating parabolic rational maps with Hecke groups

We prove that any degree $d$ rational map having a parabolic fixed point of multiplier $1$ with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group $H_{d+1}$, with the mating realized by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made in \cite{BF1}. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering.

math.DS↗

arXiv2026

Universality of the Basilica

We establish universality of the fat Basilica Julia set $J(z^2-\frac34)$ in conformal dynamics in the following sense: $J(z^2-\frac34)$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial as well as to the limit set of any geometrically finite closed surface Bers boundary group. We thus obtain the first example of a connected rational Julia set, not homeomorphic to the circle or the sphere, that is quasiconformally equivalent to a Kleinian limit set. It follows that any geometrically finite Bers boundary limit set is conformally removable. Other consequences of this universality result include quasi-symmetric uniformization of polynomial fat Basilicas by round Basilicas, and the existence of infinitely many non-commensurable uniformly quasi-symmetric surface subgroups of the Basilica quasi-symmetry group. We apply our techniques to cuspidal Basilica Julia sets arising from Schwarz reflections and cubic polynomials, yielding further universality classes. We also show that the standard Basilica Julia set $J(z^2-1)$ is the archbasilica in the David hierarchy.

math.DS↗

arXiv2025

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

We present an overview of the rapidly evolving field of dynamics of algebraic correspondences, with a focus on matings between rational maps and Kleinian groups. These correspondences exhibit rich dynamics, both within the Sullivan dictionary and beyond. We highlight unifying structures in their parameter spaces, showing how moduli spaces of rational maps and Kleinian groups naturally connect. We also outline a range of applications of the techniques developed in this framework and conclude with several promising directions.

math.DS↗

arXiv2025

On topology and singularities of quadrature domains

We prove a linear upper bound for the number of singular points on the boundary of a quadrature domain, improving a previously known quadratic bound due to Gustafsson \cite{Gus88}. This linear upper bound on the number of boundary double points also strengthens the bound on the connectivity (i.e., the number of complementary components) of a quadrature domain given by Lee and Makarov \cite{LM16}. Our proofs use conformal dynamics and hyperbolic geometry arguments. Finally, we introduce a new dynamical method to construct multiply connected quadrature domains.

math.CV↗

arXiv2025

David extension of circle homeomorphisms, welding, mating, and removability

We provide a David extension result for circle homeomorphisms conjugating two dynamical systems such that parabolic periodic points go to parabolic periodic points, but hyperbolic points can go to parabolics as well. We use this result, in particular, to prove the existence of a new class of welding homeomorphisms, to establish an explicit dynamical connection between critically fixed anti-rational maps and kissing reflection groups, to show conformal removability of the Julia sets of geometrically finite polynomials and of the limit sets of necklace reflection groups, to produce matings of anti-polynomials and necklace reflection groups, and to give a new proof of the existence of Suffridge polynomials (extremal points in certain spaces of univalent maps).

math.DS↗

arXiv2025

Simultaneous Uniformization and Algebraic Correspondences

We prove a generalization of Bers' simultaneous uniformization theorem in the world of algebraic correspondences. More precisely, we construct algebraic correspondences that simultaneously uniformize a pair of non-homeomorphic genus zero orbifolds. We also present a complex-analytic realization of the Teichmüller space of a punctured sphere in the space of correspondences.

math.GT↗

arXiv2025

Matings, holomorphic correspondences, and a Bers slice

There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: the algebraic correspondence framework due to Bullett-Penrose-Lomonaco \cite{BP94,BL20} and the simultaneous uniformization mating framework of \cite{MM23a}. The current paper unifies and generalizes these two frameworks. To achieve this, we extend the mating framework of \cite{MM23a} to genus zero hyperbolic orbifolds with at most one orbifold point of order $ν\geq 3$ and at most one orbifold point of order two. We give an explicit description of the resulting conformal matings in terms of uniformizing rational maps. Using these rational maps, we construct correspondences that are matings of such hyperbolic orbifold groups (including punctured spheres and Hecke groups) with polynomials in real-symmetric hyperbolic components. We also define an algebraic parameter space of correspondences and construct an analog of a Bers slice of the above orbifolds in this parameter space.

math.DS↗

arXiv2025

Antiholomorphic correspondences and mating II: Shabat polynomial slices

We study natural one-parameter families of antiholomorphic correspondences arising from univalent restrictions of Shabat polynomials, indexed by rooted dessin d'enfants. We prove that the parameter spaces are topological quadrilaterals, giving a partial description of the univalency loci for the uniformizing Shabat polynomials. We show that the escape loci of our parameter spaces are naturally (real-analytically) uniformized by disks. We proceed with designing a puzzle structure (dual to the indexing dessin) for non-renormalizable maps, yielding combinatorial rigidity in these classes. Then we develop a renormalization theory for pinched (anti-)polynomial-like maps in order to describe all combinatorial Multibrot and Multicorn copies contained in our connectedness loci (a curious feature of these parameter spaces is the presence of multiple period one copies). Finally, we construct locally connected combinatorial models for the connectedness loci into which the indexing dessins naturally embed.

math.DS↗

arXiv2025

Actin driven morphogenesis in hydra

Hydra, a centimeter long cylindrical-shaped freshwater organism, has emerged as an interesting model system for studying morphogenesis in animals. Recently, fluorescent imaging of cytoskeletal actin filaments on the outer surface of hydra has revealed nematic-type arrangement of actin filaments. {Several topological defects in the nematic field have also been detected. In particular, aster-like +1 defects appear at the curved head of hydra and at the tip of its tentacles, while -1/2 defects are seen at the base of the tentacles. However, functional role of these defects in tissue development is not clear. Motivated by these observations, we here model hydra's epthelial tissue as a visco-elastic membrane and the tentacles as growing membrane tubes driven by a nematic interaction among actin. We consider the epithelial layer of hydra as a fluid membrane and carry out a non-equilibrium simulation which also includes membrane growth and polymerization of actin. We show that specific kind of defect at the head does not play any positive role in emergence of the tentacles. The reorganization of actin at the base and the tip of growing tentacles are consistent with other possible defect structures at the head as well. While it is known that regions of tentacle growth are hot spots of chemical signaling, involving Wnt3/$β$-catenin pathway, we propose that active polymerization of actin bundles could also be an important player in the growth of tubular tentacles. In addition to polymerization, fluidity of our model membrane, capturing effective fluidity of the epithelial tissue, turns out to be essential for enabling such growth.

cond-mat.soft↗

arXiv2025

Correspondences on hyperelliptic surfaces, combination theorems, and Hurwitz spaces

We construct a general class of correspondences on hyperelliptic Riemann surfaces of arbitrary genus that combine finitely many Fuchsian genus zero orbifold groups and Blaschke products. As an intermediate step, we first construct analytic combinations of these objects as partially defined maps on the Riemann sphere. We then give an algebraic characterization of these analytic combinations in terms of hyperelliptic involutions and meromorphic maps on compact Riemann surfaces. These involutions and meromorphic maps, in turn, give rise to the desired correspondences. The moduli space of such correspondences can be identified with a product of Teichmüller spaces and Blaschke spaces. The explicit description of the correspondences then allows us to construct a dynamically natural injection of this product space into appropriate Hurwitz spaces.

math.DS↗

arXiv2024

Antiholomorphic correspondences and mating I: realization theorems

In this paper, we bring together four different branches of antiholomorphic dynamics: of global anti-rational maps, reflection groups, Schwarz reflections in quadrature domains, and antiholomorphic correspondences. We establish the first general realization theorems for bi-degree $d$:$d$ correspondences on the Riemann sphere (for $d\geq 2$) as matings of maps and groups. To achieve this, we introduce and study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches. Such correspondences are closely related to a class of single-valued antiholomorphic maps in one complex variable; namely, Schwarz reflection maps of simply connected quadrature domains. Using this connection, we prove that matings of all parabolic antiholomorphic rational maps with connected Julia sets (of arbitrary degree) and antiholomorphic analogues of Hecke groups can be realized as such correspondences. We also draw the same conclusion when parabolic maps are replaced with critically non-recurrent antiholomorphic polynomials with connected Julia sets.

math.DS↗

arXiv2024

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

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.

math.DS↗