Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.

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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.


ims20-05
Araceli Bonifant, John Milnor, Scott Sutherland
The W. Thurston Algorithm Applied to Real Polynomial Maps
Abstract:

This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.

  arXiv:2005.07800
Caroline Davis, Jasmine Powell, Rebecca R. Winarski, Jonguk Yang
Elastic Graphs for Main Molecule Matings
Abstract:

Recent work of Dylan Thurston gives a condition for when a post-critically finite branched self-cover of the sphere is equivalent to a rational map. We apply D. Thurston's positive criterion for rationality to give a new proof of a theorem of Rees, Shishikura, and Tan about the mateability of quadratic polynomials when one polynomial is in the main molecule. These methods may be a step in understanding the mateability of higher degree post-critically finite polynomials and demonstrate how to apply the positive criterion to classical problems.

  arXiv:2010.11382
ims20-07
Araceli Bonifant, John Milnor, Scott Sutherland
The W. Thurston Algorithm for Real Quadratic Rational Maps
Abstract:

A study of real quadratic maps with real critical points, emphasizing the effective construction of critically finite maps with specified combinatorics. We discuss the behavior of the Thurston algorithm in obstructed cases, and in one exceptional badly behaved case, and provide a new description of the appropriate moduli spaces. There is also an application to topological entropy.

arXiv:2009.10147

Yair Minsky, Babak Modami
Bottlenecks for Weil-Petersson geodesics
Abstract:

We introduce a method for constructing Weil-Petersson (WP) geodesics with certain behavior in the Teichmüller space. This allows us to study the itinerary of geodesics among the strata of the WP completion and its relation to subsurface projection coefficients of their end invariants. As an application we demonstrate the disparity between short curves in the universal curve over a WP geodesic and those of the associated hyperbolic 3-manifold.

arXiv:2001.11476

Mikhail Lyubich, Sergei Merenkov, Sabyasachi Mukherjee, Dimitrios Ntalampekos
David extension of circle homeomorphisms, welding, mating, and removability
Abstract:

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)

arXiv:2010.11256

ims20-10
Nguyen-Bac Dang, Rostislav Grigorchuk, Mikhail Lyubich
Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum
Abstract:

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of discrete spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

  arXiv:2010.00675
ims20-11
Misha Lyubich, John W. Robertson
The Critical Locus and Rigidity of Foliations of Complex Henon Maps
Abstract:

We study Henon maps which are perturbations of a hyperbolic polynomial p with connected Julia set. We give a complete description of the critical locus of these maps. In particular, we show that for each critical point c of p, there is a primary component of the critical locus asymptotic to the line y = c. Moreover, primary components are conformally equivalent to the punctured disk, and their orbits cover the whole critical set. We also describe the holonomy maps from such a component to itself along the leaves of two natural foliations. Finally, we show that a quadratic Henon map taken along with the natural pair of foliations, is a rigid object, in the sense that a conjugacy between two such maps respecting the foliations is a holomorphic or antiholomorphic affine map.

  arXiv:2101.12148
ims20-12
Jonguk Yang
Local Connectivity of Polynomial Julia sets at Bounded Type Siegel Boundaries
Abstract:

Consider a polynomial f of degree d≥2 whose Julia set Jf is connected. If f has a Siegel disc Δf of bounded type rotation number, then Jf is locally connected at the Siegel boundary ∂Δf.

arXiv:2010.14003v2

ims19-05
Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, Sabyasachi Mukherjee
Schwarz reflections and anti-holomorphic correspondences
Abstract:

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in \cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic anti-rational maps with the abstract modular group. We further illustrate our mating framework by studying the correspondence associated with the Schwarz reflection map of a deltoid.

arXiv:1907.09107v2

ims19-06
Dzmitry Dudko, Mikhail Lyubich
Local connectivity of the Mandelbrot set at some satellite parameters of bounded type
Abstract:

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.

arXiv:1808.10425v3

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