James T. Rogers
Let $f$ be a polynomial map of the Riemann sphere of degree at least two. We prove that if $f$ has a Siegel disk $G$ on which the rotation number satisfies a diophantine condition, then the boundary of $G$ contains a critical point.
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We construct holomorphic maps with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a subset of C.
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.
In this note we construct hyperbolic fixed points for cylinder renormalization of maps with Siegel disks.
We prove that for typical rotation numbers polynomial Siegel disks are Jordan domains with boundaries containing at least one critical point.
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.
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 transfer the lower semi-continuity of Siegel disks with fixed Brjuno type rotation numbers to geometric limits. Here, we restrict to Lavaurs maps associated to quadratic polynomials with a parabolic fixed point.
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.
We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative Hénon-like ma…
Local Connectivity of Polynomial Julia sets at Bounded Type Siegel Boundaries
本文证明具有有界类型旋转数的 Siegel 盘不能被刺猬包含，并进一步证明若 Julia 集连通，则其在 Siegel 边界处局部连通。
We prove here one of the claims of the author's thesis.
We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.
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.
Recently, Xavier Buff and Arnaud Cheritat have provided an elegant proof of the existence of quadratic Siegel disks with smooth boundary. In this short note, we show how results of Yoccoz and Risler can be used to conclude the same result. Our proof is a small modification of the argument given by Buff and Cheritat.
Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk such that f restricts to a homeomorphism of the boundary. We show that the Siegel disk is bounded. Using a result of Herman, we deduce that if additionally the rotation number of the Siegel disk is Diophantine, then its boundary contains a critical point of f. Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk $U$ whose boundary contains no singular values, then the condition that f is a homeomorphism of the boundary of U is automat…
We study one of the central open questions in one-dimensional renormalization theory -- the conjectural universality of golden-mean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second author. Numerical implementation of this approach relies on the Constructive Measurable Riemann Mapping Theorem proved by the first author. Our numerical study yields a convincing evidence to support the Hyperbolicity Conjecture in this setting.
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。
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。
We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.
Boundaries of Siegel Disks for Conservative Systems
本文研究了一个特定的保守标准映射在复维数2中的情况，通过数值方法可视化和分析其边界的光滑性，并提出并支持了一些猜想。
Let $f$ be an entire transcendental function of finite order and $Δ$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded orbit, then the boundary of $Δ$ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.
We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan conjecture for bounded type Siegel disks.
We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at $0$, to be parameterized by the rotation number $α$, to depend on $α$ in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.
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。
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.
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…
We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a forward invariant Siegel disk is unbounded if and only if it contains at least one asymptotic value on the boundary. Our argument is elementary and function-theoretic. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.
Cubic polynomials with a 2-cycle of Siegel disks
本文研究具有Siegel盘2-循环的三次多项式的动力学模空间，证明在旋转数有界型条件下，两个临界点均位于Siegel盘边界上的轨迹由两条弧线和一条Jordan曲线组成，分别对应临界点位于同一或不同Siegel盘边界的情形。
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…
Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points
证明对于所有次数 d≥2 和有界型无理数 θ，在具有旋转数 θ 的周期 1 Siegel 圆盘 Δ 的首一多项式空间中，Δ 关于其不动点的共形半径的最大值轨迹包含所有临界点位于 Δ 边界上的多项式，并将 Douady 猜想简化为一个更弱的陈述。
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。
If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between $0$ and $18$. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type formulas and related Grothendieck residues.
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.
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。
We give a positive answer to a question of Herman, Baker and Rippon by showing that, in the exponential family, the boundary of every unbounded Siegel disk contains the singular value.
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 investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasiperiodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this "linearization" (or conjugacy) from knowledge of a single quasiperiodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian Problem: de…
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.
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 …
Local connectivity of Julia sets of some transcendental entire functions with Siegel disks
本文利用拟Blaschke乘积长迭代的弱扩张性质，证明了一类具有有界型Siegel盘的超越整函数Julia集局部连通，并给出正弦函数实例及含渐近值情形的存在性结果。
In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels which have no "free critical points". We explain how the idea is used in solving three of the questions posed by Petersen. To present the details of the idea, we focus on the solution of one of them: we prove that for typical rotation numbers $0< θ< 1$, the boundary of the Siegel disk of $f_θ(z) = e^{2 πi θ} \sin (z)$ is a Jordan curve which passes through exac…
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。
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.
Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there …
We study the slices of the parameter space of cubic polynomials where we fix the multiplier of a fixed point to some value $λ$. The main object of interest here is the radius of convergence of the linearizing parametrization. The opposite of its logarithm turns out to be a sub-harmonic function of the parameter whose Laplacian $μ_λ$ is of particular interest. We relate its support to the Zakeri curve in the case the multiplier is neutral with a bounded type irrational rotation number. In the attracting case, we define and study an analogue of the Zakeri curve, using work of Petersen and Tan. I…
Cubic Siegel polynomials and the bifurcation measure
本文证明具有固定 Siegel 圆盘且临界轨道最终落入其中的三次多项式位于分支测度支集中，肯定回答 Dujardin 的问题，并推出支集中存在全纯圆盘、刚性参数集非闭。
Let f_n be a sequence of analytic functions in a domain U with a common attracting fixed point z_0. Suppose that f_n converges to f_0 uniformly on each compact subset of U and that z_0 is a Siegel point of f_0. We establish a sufficient condition for the immediate basins of attraction $\mathcal A^*(z_0,f_n,U)$ to form a sequence that converges to the Siegel disk of f_0 as to the kernel with respect to z_0. The same condition is shown to imply convergence of the Koenings functions associated with f_n to that of f_0. Our method allows us also to obtain a kind of quantitative result for analytic …
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。
A cubic polynomial $f$ with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call $f$ a \emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials and show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by de…
The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. 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 literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization - a novel type of renormalization for holomorphic maps with a Siegel disk w…
It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Hénon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the g…
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。
In this note we show that the regular part of the natural extension (in the sense of Lyubich and Minsky) of quadratic map $f(z) = e^{2 πi θ}z + z^2$ with irrational $θ$ of bounded type has only parabolic leaves except the invariant lift of the Siegel disk. We also show that though the natural extension of a rational function with a Cremer fixed point has a continuum of irregular points, it can not supply enough singularity to apply the Gross star theorem to find hyperbolic leaves.
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.
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。
We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting.
Rigidity of bounded-type Siegel polynomials
本文研究了具有有界类型塞格尔盘的多项式刚性，证明了其朱利亚集的局部连通性和拟共形刚性，并验证了组合刚性猜想。
Wandering domains arising from Lavaurs maps with Siegel disks
本文构造了由带 Siegel 型不动点的 Lavaurs 映射产生的新型高维游荡域，并通过抛物曲线与精确收敛速率估计证明捕获域存在的充要条件。
We prove that, for polynomials, the boundary of any bounded Fatou component is a Jordan curve, except maybe for Siegel disks.
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。
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