Research
My research sits at the intersection of complex analysis, hyperbolic geometry, and low-dimensional topology. The central objects are holomorphic and antiholomorphic dynamical systems — maps on the Riemann sphere whose iteration produces intricate fractal geometry. A recurring theme is the Fatou-Sullivan dictionary, an analogy between the theory of rational maps (iteration of polynomials and rational functions) and Kleinian groups (discrete groups of Möbius transformations). My work extends and makes this dictionary precise in the setting of Schwarz reflection maps and antiholomorphic correspondences, which simultaneously generalize both sides of the dictionary. The techniques draw on quasiconformal geometry, Teichmüller theory, combinatorial methods (dessins d’enfants), and renormalization theory.
Published in Arxiv, 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.
Recommended citation: M. Lyubich, J. Mazor, S. Mukherjee. (2024). " Antiholomorphic correspondences and mating II: Shabat polynomial slices." Arxiv:2509.12357 https://arxiv.org/abs/2509.12357
Published in Communications of the AMS, 2024
In this paper, we study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches, on the Riemann sphere. 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.
Recommended citation: M. Lyubich, J. Mazor, S. Mukherjee. (2024). " Antiholomorphic correspondences and mating I: realization theorems." Comm. of the AMS 4 (2024), 495-547. https://www.ams.org/journals/cams/2024-04-11/S2692-3688-2024-00037-8/
Published in Stony Brook University (PhD Thesis), 2024
PhD thesis. We study one-parameter families of Schwarz reflection maps of the Riemann sphere whose critical values are determined by Shabat-Belyi maps. These families arise naturally at the interface of holomorphic dynamics and the theory of dessins d’enfants. We analyze the structure of the parameter spaces, the geometry of the associated limit sets, and the relationship between Schwarz reflection maps and antiholomorphic correspondences, contributing to the Fatou-Sullivan dictionary between rational maps and Kleinian groups.
Recommended citation: J. Mazor. (2024). "One-parameter families of Schwarz reflection maps arising from Shabat-Belyi maps." PhD Thesis, Stony Brook University. https://www.math.stonybrook.edu/alumni/2024-Jacob-Mazor.pdf
