Michael Yampolsky, Saeed Zakeri
Let F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers theta and nu. Using a new degree 3 Blaschke product model for the dynamics of F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that F can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers theta and nu.
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Mating Siegel and Thurston quadratic polynomials
本文证明了有界型Siegel盘的二次多项式与后临界有限二次多项式总是可配对的，给出了复动力系统中配对定理的新结果。
Consider a quadratic polynomial with a fixed Siegel disc of bounded type. Using an adaptation of complex a priori bounds for critical circle maps, we prove that this Siegel polynomial is conformally mateable with the basilica polynomial.
Mating Siegel and parabolic quadratic polynomials
本文证明了有界型无理数和有理数对应的二次多项式共形可配对且在共轭下唯一，并据此证明了一类带抛物点的Siegel有理映射的Julia集局部连通。
In this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1/2-limb of M, is non-renormalizable, and does not have any non-repelling periodic orbits.
Mating is an operation that identifies the domains of a polynomial pair in order to obtain a new map on the resulting quotient space. The dynamics of the mating are then dependent on the two polynomials and the manner in which the quotient space was defined, which can be difficult to visualize. This research addresses using Hubbard trees and finite subdivision rules as tools to examine quadratic matings with preperiodic critical points. In many cases, discrete parameter information on such quadratic pairs can be translated into topological information on the dynamics of their mating. The centr…
In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components.
A topological mating is a map defined by gluing together the filled Julia sets of two quadratic polynomials. The identifications are visualized and understood by pinching ray-equivalence classes of the formal mating. For postcritically finite polynomials in non-conjugate limbs of the Mandelbrot set, classical results construct the geometric mating from the formal mating. Here families of examples are discussed, such that all ray-equivalence classes are uniformly bounded trees. Thus the topological mating is obtained directly in geometrically finite and infinite cases. On the other hand, renorm…
In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $\mathcal{F}_a$: $$\left(\frac{aw-1}{w-1}\right)^2+\left(\frac{aw-1}{w-1}\right)\left(\frac{az+1}{z+1}\right) +\left(\frac{az+1}{z+1}\right)^2=3$$ and proved that for every value of $a \in [4,7] \subset \mathbb{R}$ the correspondence $\mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,\,\,c \in \mathbb{R}$ and the modular group $Γ=PSL(2,\mathbb{Z})$. They conjectured that this is the case for every member of the family $\mathcal{F}_a$ which has $a$ in the co…
Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichmüller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless the orbifold of f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of p…
Topological mating is an combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $F$ on the Riemann sphere. Given a pair of polynomials of the form $z^2+c$ that are postcritically finite, there is a fast test on the constant parameters to determine whether this map $F$ exists---but this test is not constructive. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, $F$. Thi…
We develop dynamical theory for the family of holomorphic correspondences $\mathcal{F}_a$ proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice $Per_1(1)$ (in 'Mating quadratic maps with the modular group II'). Such a mating endows the complement of the limit set of $\mathcal{F}_a$ with the geometry of the hyperbolic plane, equipped with the action of the modular group. We introduce bi-infinite coding sequences for geodesics in this complement, utilising continued fraction expressions of end points; we prove landing theorems for p…
We prove here one of the claims of the author's thesis.
The Medusa Algorithm for Polynomial Matings
本文介绍美杜莎算法，用于计算两个后临界有限二次多项式的配对有理映射，并讨论其实现、收敛性判定及若干实例图像。
For certain classes of holomorphic correspondences which are matings between Kleinian groups and polynomials, we prove the existence of pinching deformations, analogous to Maskit's deformations of Kleinian groups which pinch loxodromic elements to parabolic elements. We apply our results to establish the existence of matings between quadratic maps and the modular group, for a large class of quadratic maps, and of matings between the quadratic map $z\to z^2$ and circle-packing representations of the free product $C\_2*C\_3$ of cyclic groups of order 2 and 3.
As was recently shown by the first author and others, golden-mean Siegel disks of sufficiently dissipative complex quadratic Hénon maps are bounded by topological circles. In this paper we investigate the geometric properties of such curves, and demonstrate that they cannot be $C^1$-smooth.
Equidistribution for matings of quadratic maps with the Modular group
研究二次有理映射与模群配对的全纯对应族的渐近行为，证明非例外点的迭代像与原像及周期点均等分布，即使该族在特定意义下仅为弱模而非模。
Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $α$. Let $z$ be a biaccessible point in the Julia set of $f$. Then: 1. In the Siegel case, the orbit of $z$ must eventually hit the critical point of $f$. 2. In the Cremer case, the orbit of $z$ must eventually hit the fixed point $α$. Siegel polynomials with biaccessible critical point certainly exist, but in the Cremer case it is possible that biaccessible points can never exist. As a corollary, we conclude that the set of biaccessible points in the Julia set of a Siegel or Cremer quadratic polynomial has Bro…
In the quadratic family (the set of polynomials of degree 2), Petersen and Zakeri proved the existence of Siegel disks whose boundaries are Jordan curves, but not quasicircles. In their examples, the critical point is contained in the curve. In the first part, we prove the existence of quadratic examples that do not contain the critical point. In the second part, using a more abstract point of view (suggested by Avila), we show that we can control quite precisely the degree of regularity of the boundary of the quadratic Siegel disks we create by perturbations. For instance there exists example…
We continue the study of constructing invariant Laplacians on Julia sets, and studying properties of their spectra. In this paper we focus on two types of examples: 1) Julia sets of cubic polynomials $z^3 + c$ with a single critical point; 2) formal matings of quadratic Julia sets. The general scheme introduced in earlier papers in this series involves realizing the Julia set as a circle with identifications, and attempting to obtain the Laplacian as a renormalized limit of graph Laplacians on graphs derived form the circle with identifications model. In the case of cubic Julia sets the detail…
On quadratic Siegel disks with a class of unbounded type rotation numbers
本文研究具有无界型旋转数Siegel盘的二次多项式，证明其Siegel盘边界点均为填充Julia集的Lebesgue密度点，推广了McMullen的有界型结果，还验证了拟共形手术标准候选模型的相关密度点性质，可用于构造更多正面积二次Julia集。
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.
The high type quadratic Siegel disks are Jordan domains
本文证明高型二次 Siegel 圆盘的边界为 Jordan 曲线，并刻画其包含临界点当且仅当旋转数为 Herman 数。
On the area rate of perturbed Siegel disks
本文研究了扰动Siegel盘的面积率，证明了在构造具有正面积的二次Julia集时，Buff和Chéritat对旋转数添加的限制性条件可以被删除，从而能够构造出更多此类Julia集。
If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2iπ\a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a …
We show the existence of a rational surface automorphism of positive entropy with a given number of Siegel disks. Moreover, among automorphisms obtained from quadratic birational maps on the projective plane fixing irreducible cubic curves, we find out an automorphism of positive entropy with multiple Siegel disks.
For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.
Uniform a priori bounds for neutral renormalization
本文证明有界型Siegel盘的均匀伪Siegel先验界，构造具有星状结构的母刺猬以控制二次多项式后临界集，并将Siegel/Pacman与近抛物重整化理论推广到所有近中性二次多项式。
If alpha is an irrational number, we define Yoccoz's Brjuno function Phi by Phi(alpha)=sum_{n geq 0} alpha_0*alpha_1*...*alpha_{n-1}*log(1/alpha_n), where alpha_0 is the fractional part of alpha and alpha_{n+1} is the fractional part of 1/alpha_n. The numbers alpha such that Phi(alpha)<infty are called the Brjuno numbers. The quadratic polynomial P_alpha:z -> e^{2i pi alpha}z+z^2 has an indifferent fixed point at the origin. If P_alpha is linearizable, we let r(alpha) be the conformal radius of the Siegel disk and we set r(alpha)=0 otherwise. Yoccoz proved that Phi(alpha)=infty if and only if …
Mating quadratic maps with the modular group III: The modular Mandelbrot set
本研究证明Bullett-Penrose(2:2)全纯对应族的连通性轨迹与抛物Mandelbrot集间存在动力学同胚，结合已有结论推导出该轨迹与经典Mandelbrot集同胚，完善了二次映射与模群配对的相关理论。
Rigidity of bounded type cubic Siegel polynomials
本文证明有界型旋转数的不可重整化三次Siegel多项式的组合等价蕴含共形等价，并由此验证了Blokh-Oversteegen-Ptacek-Timorin关于Zakeri单参数切片中不可重整化映射轨迹拓扑结构的猜想。
Let $f(z)=e^{2iπθ} z+z^2$, where $θ$ is a quadratic irrational. McMullen proved that the Siegel disk for $f$ is self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if $θ=(\sqrt 5-1)/2$ is the golden mean, then there exists a triangle contained in the Siegel disk, and with one vertex at the critical point. This answers a 15 year old conjecture.
Uniform a priori bounds for neutral renormalization. Variation II: $ψ^\bullet$-ql Siegel maps
本研究将中性二次多项式的一致伪Siegel界推广至$ψ^\ullet$-类二次Siegel映射，证明Siegel盘退化在组合区间上等分布，精确刻画了该结构除过渡标度外各几何标度下的退化特征，且界可迁移至多类有理映射族。
For quadratic polynomials of one complex variable, the boundary of the golden-mean Siegel disk must be a quasicircle. We show that the analogous statement is not true for quadratic Hénon maps of two complex variables.
We present an explicit formula relating volumes of strata of meromorphicquadratic differentials with at most simple poles on Riemann surfacesand counting functions of the number of flat cylinders filled by closedgeodesics in associated flat metric with singularities. This generalizes the resultof Athreya, Eskin and Zorich in genus 0 to higher genera.
The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling ratio. The restriction of the dynamics of the quadratic polynomial to the boundary of the Siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the…
Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given irrational number of Brjuno type. Our main goal is to prove that when theta is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense par…
Arithmetic conditions for Julia sets to have positive area
针对边界不含临界点的带Cremer点或Siegel圆盘的二次多项式，研究人员利用Siegel返回机制与伪Siegel圆盘理论，给出其Julia集具有正面积的算术条件。
Uniform bounds for bubbles of neutral quadratic polynomials
本文针对具有有界类型无理不动点的二次多项式，通过引入近退化机制，获得了伪气泡大小和正则性的一致界，为中性重整化的组合刚性提供了关键基础。
We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite) renormalizations.
We extend Thurston's combinatorial criterion for postcritically finite rational maps to a class of rational maps with bounded type Siegel disks. The combinatorial characterization of this class of Siegel rational maps plays a special role in the study of general Siegel rational maps. As one of the applications, we prove that for any quadratic rational map with a bounded type Siegel disk, the boundary of the Siegel disk is a quasi-circle which passes through one or both of the critical points.
Local connectivity of Julia sets of some rational maps with Siegel disks
本文证明有理映射长迭代在有界型Siegel盘边界附近扩张，借此将Petersen关于二次Siegel多项式Julia集局部连通性的结果推广至一般情形，且证明无需使用puzzles。
X. Buff and A. Cheritat proved that there are quadratic polynomials having Siegel disks with smooth boundaries. Based on a simplification of A. Avila, we give yet another simplification of their proof. The main tool used is a harmonic function introduced by Yoccoz whose boundary values are the sizes of the Siegel disks. The proof also applies to some other families of polynomials, entire and meromorphic functions.
We prove higher moment formulas for Siegel transforms defined over the space of unimodular $S$-lattices in $\mathbb Q_S^d$, $d\ge 3$, where in the real case, the formulas are introduced by Rogers (1955). As applications, we obtain the random statements of Gauss circle problem for any convex sets in $\mathbb Q_S^d$ containing the origin and of the effective Oppenheim conjecture for $S$-arithmetic quadratic forms.
It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.
Lebesgue measure of the postcritical set of neutral quadratic polynomials
本文证明所有具有中立不动点的二次多项式的后临界集Lebesgue测度为零，无需旋转数算术条件，通过扇形重整化与伪Siegel界实现，并推广至重整化塔及母刺猬内部的最优Brjuno准则。
Existence of Exotic rotation domains and Herman rings for quadratic Hénon maps
本文证明了二次Hénon映射中“Exotic rotation domains”和耗散情形下吸引性Herman环的存在性，并提供了数值生成这些Herman环的理论框架。
On postcritical sets of quadratic polynomials with a neutral fixed point
本文基于 Dudko 和 Lyubich 的伪 Siegel 盘理论，实现了具有中性不动点的二次多项式后临界集在一般情形下的控制，推广了此前仅限于特定旋转数类型的研究成果。
Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields
在函数域中证明了Oppenheim猜想的定量版本，通过计算Siegel变换的高阶矩，得到了集合中格点数的方差最优界，并精确计算了球内格点数的方差。
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geo…
We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the dynamics of some holomorphic maps of the form $f(z) = e^{2πi α} z + \mathcal{O}(z^2)$, including the quadratic polynomials $e^{2πi α} z+z^2$, for some irrational values of $α$. The main results of the paper concern fine-scale features of the measure-theoretic attractors of these maps, and their dependence on the data. As a bi-product, we establish an optimal upper bound on the size of the maximal linearization domain …
In this paper we analyze the large genus asymptotics for intersection numbers between $ψ$-classes, also called correlators, on the moduli space of stable curves. Our proofs proceed through a combinatorial analysis of the recursive relations (Virasoro constraints) that uniquely determine these correlators, together with a comparison between the coefficients in these relations with the jump probabilities of a certain asymmetric simple random walk. As an application of this result, we provide the large genus limits for Masur-Veech volumes and area Siegel-Veech constants associated with principal …
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