Saaed Zakeri
Let $f:z\mapsto z^2+c$ be a quadratic polynomial whose Julia set $J$ is locally-connected of the set of biaccessible points in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev quadratic polynomial for which the corresponding measure is one.
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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…
We give an alternative way to construct an entire function with quasiconformal surgery so that all its Fatou components are quasi-circles but the Julia set is non-locally connected.
One of the main questions in the field of complex dynamics is the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected. In this paper we shall prove the following Main Theorem: Let $f$ be a polynomial of the form $f(z)=z^d +c$ with $d$ an even integer and $c$ real. Then the Julia set of $f$ is either totally disconnected or locally connected. In particular, the Julia set of $z^2+c$ is locally connected if $c \in [-2,1/4]$ and totally disconnected otherwise.
According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this article, we reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point which eventually maps to …
Most Fatou and Julia components are small for polynomials
本文证明了多项式的 Julia 分量在直径上通常很小，且对于无无理性中性循环的多项式，即使 Julia 集非局部连通，其 Fatou 分量通常也很小。
We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.
We show that if $β>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^β+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of the associated pressure function. As a consequence, we conclude that the Julia set of the correspondence has zero Lebesgue measure for parameters close to zero, whenever $q^2<p$ and $β=p/q$ in lowest terms.
We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet-Eckmann and Topological Collet-Eckmann maps and maps verifying a summability condition (as considered by Graczyk and Smirnov).
We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in math.DS/0505191 that give control of moduli of annuli under maps of high degree.
In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.
Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in the general (not necessarily hyperbolic) case.
Non-locally connected spaces I: Coverings and branched coverings on metric spaces
研究一般度量空间上覆盖与分支覆盖，通过提升连续统发展理论及替代同伦理论，推广经典结果，并研究其在球面分支覆盖动力学中的应用及意义。
The following notes provide an introduction to recent work of Branner, Hubbard and Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992. I am indebted to help from the audience. Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets. It presents only the "easy" part of his work, in the sense that it considers only non-renormalizable polynomials, and makes no effort to describe the much more difficult arguments which are needed to deal with local connectivity in parameter space. …
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 study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points…
Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a \emph{finest monotone map $\ph$ onto a locally connected continuum $J_{\sim_P}$}, i.e. a monotone map $\ph:J\to J_{\sim_P}$ such that for any other monotone map $ψ:J\to J'$ there exists a monotone map $h$ with $ψ=h\circ \ph$. Then we extend $\ph$ onto the complex plane $\C$ (keeping the same notation) and show that $\ph$ monotonically semiconjugates $P|_{\C}$ to a \emph{topological polynomial $g:\C\to \C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proo…
Fixed Flowers
本文利用lamination研究Julia集，探讨了不动点画像的全局计数，以及局部单临界lamination与带旋转多边形的局部最大临界lamination之间的对应关系。
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。
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.
We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of fibers of Mandelbrot and Multibrot sets is preserved under tuning maps and other (partial) homeomorphisms. Similarly, we show for unicritical polynomials that triviality of fibers of …
Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected continuum (i.e., $m^{-1}(y)$ is connected for each $y\in L$, and any monotone map $φ:X\to L'$ onto a locally connected continuum is a composition $φ=φ'\circ m$ where $φ':L\to L'$ is monotone). Such finest locally connected model $L$ of $X$ is easier to understand because $L$ is locally connected (in particular it can be described by a picture) and represents…
Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Set
该研究针对二次族的非逃逸轨迹$\text{KM}$，证明其边界具有满豪斯多夫维数4，在双曲参数处局部连通，在部分实抛物参数处局部连通性不成立，还构造混合谜题片分析其局部连通性。
For an infinitely renormalizable quadratic map $f_c: z\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\limsup k_m^{-1}\log |p_m|>0$, then the Mandelbrot set is locally connected at $c$. We prove also that if $\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\to \infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed…
We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and non-atomic conformal measure with exponent equal to the Hausdorff dimension of the Julia set. There is an absolutely continuous invariant measure with respect to this conformal measure. The Minkowski dimension of the Julia set is strictly less than 2. Either the Julia set is the whole sphere, or the dynamics is unstable. For such polynomials and Blaschke pr…
We show that if $P$ is a quadratic polynomial with a fixed Cremer point and Julia set $J$, then for any monotone map $\ph:J\to A$ from $J$ onto a locally connected continuum $A$, $A$ is a single point.
Parabolic Julia Sets are Polynomial Time Computable
本文证明了有理函数的抛物线Julia集在多项式时间内是局部可计算的。
We will show the Mandelbrot set $M$ is locally conformally inhomogeneous: the only conformal map $f$ defined in an open set $U$ intersecting $\partial M$ and satisfying $f(U\cap\partial M)\subset \partial M$ is the identity map. The proof uses the study of local conformal symmetries of the Julia sets of polynomials: we will show in many cases, the dynamics can be recovered from the local conformal structure of the Julia sets.
On locally distinguishing Sierpiński dynamical systems
证明在凸余紧Kleinian群的极限集和后临界有限有理映射的Julia集均为Sierpiński地毯时，两者之间不存在局部拟共形映射。
The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou-Julia-Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.
Newton's method applied to rational functions: Fixed points and Julia sets
本文研究了牛顿映射在有理函数中的固定点与朱利亚集性质，证明了特定条件下朱利亚集的连通性和局部连通性。
A model for planar compacta and rational Julia sets
该研究提出平面紧集的具有Peano商的核心分解与原子概念，证明分支覆盖下紧集原子的原像分支仍为原像紧集的原子，将多项式下完全不变紧集的相关结果推广到有理函数情形。
Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.
In this paper, the main focus is on the Sierpinski carpet Julia sets of the rational maps with non-recurrent critical points. We study the uniform quasicircle property of the peripheral circles, the relatively separated property of the peripheral circles and the locally porous property of these carpets. We also establish some quasisymmetric rigidities of these carpets, which generalizes the main results of Bonk-Lyubich-Merenkov to the postcritically infinite case. In the end we give a strategy to construct a class of postcritically infinite rational maps whose Julia sets are quasisymetrically …
In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to a question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.
Uniformization of gasket Julia sets
本文刻画了可由圆形垫片一致化的有理映射垫片Julia集，证明了胖垫片Julia集可拟共形一致化，拟圆盘Fatou分支的Julia集可通过David同胚一致化，且垫片Julia集与Kleinian群极限集可局部拟共形同胚。
Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of al…
A Peano continuum means a locally connected continuum. A compact metric space is called a \emph{Peano compactum} if all its components are Peano continua and if for any constant $C>0$ all but finitely many of its components are of diameter less than $C$. Given a compact set $K\subset\mathbb{C}$, there usually exist several upper semi-continuous decompositions of $K$ into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the \emph{core decomposition of $K$ with P…
We give a topological description of the space of quadratic rational maps with superattractive two-cycles: its "non-escape locus" M2 (the analog of the Mandelbrot set M) is locally connected, it is the continuous image of M under a canonical map, and it can be described as M (minus the 1/2-limb), mated with the lamination of the basilica. The latter statement is a refined version of a conjecture of Ben Wittner, which in its original version requires local connectivity of M to even be stated. Our methods of mating with a lamination also apply to dynamical matings of certain non-locally connecte…
Quasisymmetries of finitely ramified Julia sets
本文建立有限分支分形的拟对称性与无畸变度量理论，解决拓扑刚性分形的拟对称一致化问题，并证明多类有限分支 Julia 集具有无限拟对称群或包含 Thompson 群 $F$。
Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The \emph{red dwarf} $J_P$ are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing $p$ and the orbits of all critical images. The \emph{solar} $J_P$ are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and $J_P$ is connected im kleinen at its landing point. We study bi-accessible points and loca…
We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like $n^{5 + ε}$, for some $ε> 0$, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with "small" multipliers. Somehow surprinsingly the proof is based in measure theorical considerations.
Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6
本文研究仿射三次曲面自同构群的动力学，定义并分析其Julia与Fatou集，证明Julia集中存在稠密轨道，并揭示Fatou/Julia二分法与局部离散性二分法的关联，为Painlevé 6方程及特征簇研究提供新视角。
We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4^n/p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected
A holomorphic endomorphism f of CP^2 admits a Julia set J_1, defined as usual to be the locus of non-normality of its iterates, and a (typically) smaller Julia set J_2, which is essentially the closure of the set of repelling periodic orbits. The question has been raised whether J_1\J_2 is filled (possibly in a measure-theoretic sense) with "Fatou subvarieties" along which the dynamics is locally equicontinuous. In this article we construct examples showing that this is not the case in general
A compactum $X\subset \C$ is unshielded if it coincides with the boundary of the unbounded component of $\C\sm X$. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum $X$ admits a topologically unique monotone map $m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $X$ onto a finitely Suslinian quotient factors through $m_X$. We call the pair $(X_{FS},m_X)$ (or, more loosely, $X_{FS}$) the finest finitely Suslinian mo…
Geometrically finite transcendental entire functions
本文引入“温顺函数”概念刻画超越整函数的拓扑动力学，证明一大类具有界临界性的几何有限超越整函数满足该性质，类比了多项式Julia集局部连通性的经典结果并推广了前期工作。
We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set and show that fibers of certain points are ``trivial'', i.e., they consist of single points. This implies local connectivity at these points. Locally, triviality of fibers is strictly stronger than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial,…
A frequent problem in holomorphic dynamics is to prove local connectivity of Julia sets and of many points of the Mandelbrot set; local connectivity has many interesting implications. The intention of this paper is to present a new point of view for this problem: we introduce fibers of these sets, and the goal becomes to show that fibers are ``trivial'', i.e. they consist of single points. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles. This implies local connectivity at these points, but triviality of fibers is a somewhat stronger property than local connect…
A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials, even conjectural models are missing, one possible reason being that the higher degree analog of the MLC conjecture is known to be false. We provide a combinatorial model for an essential part of the parameter space of complex cubic polynomials, namely, for the spac…
Symmetry and dynamics of Chebyshev's method
本文研究切比雪夫求根方法的对称性与动力学，证明满足缩放定理时中心化多项式的对称群包含于切比雪夫映射的对称群，并刻画了等号成立的多类条件及相应朱利亚集的连通性与局部连通性。
Bialgebraic geometry of Böttcher coordinates
研究多项式 f 的 Böttcher 坐标 Ψ_f 下的双代数集，在 Julia 集不连通或连通且具有非退化局部连通模型时给出完全动力学分类，并证明类似 Ax-Lindemann-Weierstrass 定理的结论。
A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at most finitely renormalizable parameter values. One of our goals is to prove MLC for some infinitely renormalizable parameter values. Loosely speaking, we need all renormalizations to have bounded combinatorial rotation number (assumption C1) and sufficiently high combinatorial type (assumption C2). For real quadratic polynomials of bounded combinatorial type t…
Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, and assume that for any cyclic Fatou component $W$ of $f$ having a period $p\in\mathbb{N}$, the equilibrium measure $μ_f$ has a positive charge on the boundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniform a priori bound of the dynamics of $f$, which in particular yields a Diophantine-type estimate of the dynamics of $f$ on its domaines singuliers. We also point out that in the case of $k=1$, the statement of our assumption is related to both the impossibility for the Julia set of $f$ to be the bound…
Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, $d\ge 2$) and defined \emph{wandering $k$-gons} as sets $\T\subset \ucirc$ such that $\si_d^n(\T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\si_d^n(\T)$ in the plane are pairwise disjoint. He proved that $\si_2$ has no wandering $k$-gons. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. In a recent paper it was shown that uncountably many cubic WT-laminations, with pairwise non-conjugate induced maps on the corresponding …
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。
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