Parabolic Julia Sets are Polynomial Time Computable
Mark Braverman
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。
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。
In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time.
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We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.
We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
We completely characterize the conformal radii of Siegel disks in the family $$P_θ(z)=e^{2πiθ}z+z^2,$$ corresponding to {\bf computable} parameters $θ$. As a consequence, we constructively produce quadratic polynomials with {\bf non-computable} Julia sets.
We show that if a polynomial filled Julia set has empty interior, then it is computable.
We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.
We discuss computability of impressions of prime ends of compact sets. In particular, we construct quadratic Julia sets which possess explicitly described non-computable impressions.
We prove that the Julia set of a rational function $f$ is computable in polynomial time, assuming that the postcritical set of $f$ does not contain any critical points or parabolic periodic orbits.
In this note we give answers to questions posed to us by J.Milnor and M.Shub, which shed further light on the structure of non-computable Julia sets.
Julia sets appear quasiconformally in the Mandelbrot set, II: A parabolic proof
本研究遵循A. Douady的思路，给出抛物型证明，证实Mandelbrot集任意边界点附近存在嵌套于对应Julia集拟共形副本中的Mandelbrot集小型拟共形副本。
In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.
Assouad type dimensions of parabolic Julia sets
本文证明抛物型Julia集的Assouad维数为max{1, h}，其中h是Hausdorff维数，并推导了所有Assouad型维数的公式。
We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex Hénon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue $(1+t)λ$, where $λ$ is a root of unity and $t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $|t|$, the Hénon map is hyperb…
Continuity of Julia sets and invariant rays
本文研究了有理映射在抛物线扰动下的Julia集连续性与不变射线的关系，通过抛物线冲击理论建立了等价刻画。
On the parabolic Fatou domains
本文证明无限连通固定抛物型Fatou域可共形共轭为具有Jordan曲线边界和至多一个后临界点的完全不变抛物型Fatou域，且Cantor集情形下可扰动为吸引域。
Consider the parameter space $\mathcal{P}_λ\subset \mathbb{C}^{2}$ of complex Hénon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $λ=e^{2πi p/q}$. We give a characterization of those Hénon maps from the curve $\mathcal{P}_λ$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $λ$. We prove that there is an open disk of parameters in $\mathcal{P}_λ$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+…
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.
We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.
We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.
Let $d(c)$ denote the Hausdorff dimension of the Julia set $J_c$ of the polynomial $f_c(z)=z^2+c$. We will investigate behavior of the function $d(c)$ when real parameter $c$ tends to a parabolic parameter.
Parabolic Implosion in the Parameter Space of Cubic Polynomials
本文研究三次多项式参数空间中抛物内爆引起的分岔轨迹丰富化的拓扑结构，并将其与 P. Lavaurs 系统研究过的二次多项式 Julia 集丰富化建立联系。
It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \bM$, the Julia set of $z \mapsto z^2+c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.
We prove the existence of Cantor Julia sets with Hausdorff dimension two. In particular, such examples can be found in cubic polynomials. The proof is based on the characterization of the parameter spaces and dynamical planes of cubic polynomials by Branner-Hubbard, and the parabolic bifurcation theory developed by Shishikura in 1990s.
Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies
本文引入双曲和抛物二次映射填充 Julia 集的嵌铺，利用瓦片组织内部动力学，并基于嵌铺与线性化坐标退化研究构造了双曲到抛物退化的捏半共轭连续族。
We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a real parameter $c$ such that the computational complexity of computing $J_c$ with $n$ bits of precision is higher than $T(n)$. This is the first known class of real parameters with a non poly-time computable Julia set.
Using parabolic enrichment, it is shown that Gronwall area formula for the filled Julia set along the boundary of the main cardioid of the Mandelbrot set cannot be well approximated by replacing it by a finite sum. A revised version of this article will appear in the Journal of the London Mathematical Society.
Horn maps of semi-parabolic Hénon maps
本文证明了半抛物Hénon映射的Horn映射满足弱Ahlfors岛性质，从而统一了Julia集的两种定义，并证明小扰动可使前向Julia集的Hausdorff维数任意接近4。
For a class of polynomial maps of one variable with a parabolic fixed points and degrees bigger than $21$, the parabolic renormalization is introduced based on Fatou coordinates and horn maps, and a type of maps which are invariant under the parabolic renormalization is also given. For the small perturbation of these kinds of maps, the near-parabolic renormalization is also introduced based on the first return maps defined on the fundamental regions. As an application, we show the existence of non-renormalizable polynomial maps with degrees bigger than $21$ such that the Julia sets have positi…
Rational maps with Cantor bubble Julia sets
本文研究有理映射中康托尔气泡Julia集的存在条件，构造了高周期吸引周期和二维Hausdorff维数的气泡Julia集，并给出其与康托尔圆气泡准对称等价的充分条件。
It was known that the Sierpiński carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having exactly one fixed (super-) attracting or parabolic basin. We show that this criterion can be applied to some well known rational maps, such as McMullen maps and Morosawa-Pilgrim family. Moreover, we give also some special examples whose Julia sets are Sierpiński carpets.
Let $f$ be a rational map with degree $d\geq 2$ whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map $g$ such that $g$ contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of $f$ on the Julia set if and only if $f$ does not have parabolic basins and Siegel disks. If such $g$ exists, then the degree can be chosen such that $\text{deg}(g)\leq 7d-2$. In particular, if $f$ is a polynomial, then $g$ can be chosen such that $\text{deg}(g)\leq 4d+4$. Moreover, some quartic and cubic rational maps…
We prove fixed point results for branched covering maps $f$ of the plane. For complex polynomials $P$ with Julia set $J_P$ these imply that periodic cutpoints of some invariant subcontinua of $J_P$ are also cutpoints of $J_P$. We deduce that, under certain assumptions on invariant subcontinua $Q$ of $J_P$, every Riemann ray to $Q$ landing at a periodic repelling/parabolic point $x\in Q$ is isotopic to a Riemann ray to $J_P$ relative to $Q$.
Rational maps whose Julia sets are generalized Sierpiński gaskets
本文证明了包含Siegel盘、Cremer点或无穷次重正规化的有理映射也能产生广义Sierpiński垫圈Julia集，由此确立了正面积垫圈Julia集的存在性，并给出了判定其存在性的准则。
Julia sets appear quasiconformally in the Mandelbrot set
本文证明在小 Mandelbrot 集的抛物型或 Misiurewicz 参数附近可拟共形地看到 Cantor Julia 集及其嵌套装饰结构，并由此证明半双曲但非 Misiurewicz 且非双曲的参数在 Mandelbrot 集边界中稠密且 Hausdorff 维数为 2。
We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.
Sierpiński carpet Julia sets of transcendental entire functions
本文证明谢尔宾斯基地毯参数在有限型超越整函数族的双曲分支点集中稠密，并构造了具有此类 Julia 集且含吸引盆、抛物盆、Siegel 盘或游荡域的具体函数。
We prove that the meandering set for $f_a(z)=e^z+a$ is homeomorphic to the space of irrational numbers whenever $a$ belongs to the Fatou set of $f_a$. This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the radial Julia set of $f_a$ has topological dimension zero for all attracting and parabolic parameters, including all $a\in (-\infty,-1]$. Similar results are obtained for Fatou's function $f(z)=z+1+e^{-z}$.
A dynamical algorithm to compute hyperbolic Julia sets in polynomial time
本文提出一种动态算法，用于在多项式时间内计算复多项式的双曲Julia集，并确立其下可计算性。
We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycle…
Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere , such that for every $f$-critical point $c\in J$ which forward trajectory does not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast (Collet--Eckmann condition), there are no parabolic periodic points, and else such that Julia set is not the whole sphere. Then smooth (Riemann) measure of the Julia set is 0. For $f$ satisfying additionally Masato Tsujii's condition that the average distance of $f^n(c)$ from the set of critical points is not too small, we prove that Hausdorff dimension of Julia s…
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。
From Hyperbolic to Parabolic Parameters along Internal Rays
本文研究二次族沿内部射线趋近抛物参数时 Julia 集点的全纯运动，给出一致导数估计，并证明指数仅依赖花瓣数的一致单侧 Hölder 连续性。
Uniformization of Julia sets of geometrically finite rational maps
该研究刻画了Julia集与Schottky集拟共形等价的几何有限有理映射，证明了几何有限单临界多项式无穷远吸引域闭包与Schottky集拟共形等价的充要条件，并给出了拟圆周对的相关动力学刻画。
Let $d(\varepsilon)$ and $\mathcal D(δ)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_δ(z)=(1+δ)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ and $\mathcal D(δ)$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\varepsilon\in\mathbb{R}^+$ (or two imaginary directions in the $δ$ parametrization) which is outside the Mandelbrot set and is related to the paraboli…
This article focus on the connected locus of the cubic polynomial slice $Per_1(λ)$ with a parabolic fixed point of multiplier $λ=e^{2πi\frac{p}{q}}$. We first show that any parabolic component, which is a parallel notion of hyperbolic component, is a Jordan domain. Moreover, a continuum $\mathcal{K}_λ$ called the central part in the connected locus is defined. This is the natural analogue to the closure of the main hyperbolic component of $Per_1(0)$. We prove that $\mathcal{K}_λ$ is almost a double covering of the filled-in Julia set of the quadratic polynomial $P_λ(z) = λz+z^2$.
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…
Computability of Julia sets for complex Hénon maps: The role of attracting and neutral cycles
该研究将复Hénon映射Julia集的可计算性从双曲情形扩展到含吸引盆地的非双曲情形，基于鞍点稳定流形的算法实现，同时指出中性动力学下高维存在部分类似一维的可计算性二分法。
The boundaries of the hyperbolic components of odd period of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. One of the main results of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. This is achieved by constructing a complex one-dimensional quasiconformal deformation space of the parabolic arcs which are contained in the dynamically defined algebraic curves $\mathrm{Per}_n(1)$ of a suitably complexified family of polynomials. As another applicat…
Boundaries of hyperbolic and simply parabolic Baker domains
本文研究超越整函数与亚纯函数中非单叶单连通双曲型及单抛物型 Baker 域的边界，揭示其非遍历、非回归等性质，并证明边界上存在非逃逸轨道点及周期点。
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。
Brolin-Lyubich measure $λ_R$ of a rational endomorphism $R:\riem\to\riem$ with $°R\geq 2$ is the unique invariant measure of maximal entropy $h_{λ_R}=h_{\text{top}}(R)=\log d$. Its support is the Julia set $J(R)$. We demonstrate that $λ_R$ is always computable by an algorithm which has access to coefficients of $R$, even when $J(R)$ is not computable. In the case when $R$ is a polynomial, Brolin-Lyubich measure coincides with the harmonic measure of the basin of infinity. We find a sufficient condition for computability of the harmonic measure of a domain, which holds for the basin of infinity…
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。
David extension of circle homeomorphisms, welding, mating, and removability
本文通过圆周同胚的David扩张结果，在抛物与双曲周期点之间建立共轭联系，并据此证明新型焊接同胚存在性、反有理映射与反射群间的动力联系、相关Julia集与极限集的共形可移除性，以及反多项式与项链反射群的配对和Suffridge多项式的存在性。
The specification approach to equilibrium states for parabolic rational maps
本文研究抛物理性理映射平衡态的规范与轨道分解方法，扩展了已知结果，证明了在压力间隙条件下平衡态的唯一性，并展示了其K性质和正熵性。
Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps …
Computability for Axiom A Polynomial Skew Products of $\mathbb{C}^2$
本文首次研究多复变映射可计算性，针对$\text{ℂ}^2$的Axiom A多项式斜积证明其链回归集可计算，算法能识别不同类型双曲集，还得出Axiom A半可判定性及双曲性轨迹下半可计算性的结论。
Local connectivity of boundaries of tame Fatou components of meromorphic functions
本文研究超越亚纯映射不变单连通吸引盆边界的局部连通性，在“无穷远处排斥花瓣”等条件下证明结论，并应用于 Newton 方法，给出 $\mathcal B$ 类外的非平凡局部连通 Julia 集例子。
An omitted value of a transcendental meromorphic function $f$ is called a Baker omitted value, in short \textit{bov} if there is a disk $D$ centered at the bov such that each component of the boundary of $f^{-1}(D)$ is bounded. Assuming that the bov is in the Fatou set of $f$, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If $U$ is the Fatou component containing the bov then it is proved that a Fatou component $U'$ is infinitely connected if and only if it lands on $U$, i.e. $f^{k}(U') \subset U$ for some $k \geq 1…
Consider the standard family of complex Hénon maps $H(x,y) = (p(x) - ay, x)$, where $p$ is a quadratic polynomial and $a$ is a complex parameter. Let $U^{+}$ be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set $U^{+}$ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of $(\mathbb{C}-\overline{\mathbb{D}}) \times\mathbb{C}$ by a discrete group of automorphisms $Γ$ isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. On the other hand, the boundary $J^{+}$ of $U^{+}$ is a complicated fractal object on which…
Limits of Quadratic Rational Maps: The Cantor Locus
本文研究二次有理映射 Cantor 轨迹边界的抛物动力学，提出动力学标记与局部参数化，证明固定标记序列在亚循环极限下收敛于抛物参数或经重标度后趋于边界。
Chebyshev's method for exponential maps
该研究刻画了对应有理Chebyshev映射的整函数形式，分析了一类仿射共轭于$z e^{z^n}$的Chebyshev映射$C_n$的不动点、Julia集对称性与连通性等动力学性质，还确定了对应Newton方法的动力学特征。
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