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* Starred papers have appeared in the journal cited.
In this paper we prove a priori bounds for an ``elephant eye'' combinatorics. Little 
-copies specifying these combinatorics are allowed to converge to the cusp of the Mandelbrot set. To handle it, we develope a new geometric tool: uniform thin-thick decompositions for bordered Riemann surfaces.
Submitted 29 January, 2026; originally announced January 2026.
In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its geometric counterparts.
Submitted 30 December, 2025; originally announced December 2025.
We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to 
-quadratic-like Siegel maps. In this form, the bounds are compatible with the -quadratic-like renormalization theory and are easily transferable to various families of rational maps. The main theorem states that the degeneration of a Siegel disk is equidistributed among combinatorial intervals. This provides a precise description of how the -quadratic-like structure degenerates around the Siegel disk on all geometric scales except on the ``transitional scales'' between two specific combinatorial levels.
Submitted 30 September, 2025; v1 submitted 26 September, 2025; originally announced September 2025.
We present a proof of the conjecture by Bonifant and Milnor (see arXiv:2503.08868) regarding the similarity between the connectedness locus of the curve 
at Misiurewicz parameters and their corresponding filled Julia sets in a neighborhood of the corresponding free co-critical point. The proof is in parallel with the generalization of Tan Lei's proof of similarity in the Mandelbrot set developed by Kawahira.
Submitted 30 October, 2025; originally announced October 2025.
We establish certain uniform a priori bounds for hyperbolic components of disjoint type. As an application, we will prove that Sierpinski carpet hyperbolic components of disjoint type are bounded. Furthermore, we show that for each map 
on the closure of such a hyperbolic component, there exists a quadratic-like restriction around every non-repelling periodic point. Extensions of these results to non-Sierpinski configurations are underway. As a prototype example, we describe the post-critical set of any map on the boundary of the hyperbolic component of 
.
Submitted 29 September, 2025; originally announced September 2025.
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.
Submitted 25 September, 2025; originally announced September 2025.
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 space.
Submitted 26 August, 2025; originally announced August 2025.
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.
Submitted 15 September, 2025; originally announced September 2025.
We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to -quadratic-like Siegel maps. In this form, the bounds are compatible with the -quadratic-like renormalization theory and are easily transferable to various families of rational maps. The main theorem states that the degeneration of a Siegel disk is equidistributed among combinatorial intervals. This provides… ▽ More
Submitted 30 September, 2025; v1 submitted 26 September, 2025; originally announced September 2025.
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.
Submitted 17 April, 2025; originally announced April 2025.
| arXiv:2504.13107 |