Quadratic Julia Sets with Positive Area
Xavier Buff, Arnaud Cheritat
我们证明了存在的二次多项式的朱利亚集与积极的勒贝格测度在三种情况下：存在的Cremer不动点，存在的Siegel磁盘，存在无限多（卫星）重整化。
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
Biaccessiblility in quadratic Julia sets I: The locally-connected case
设$f:z\mapsto z^2+c$是一个二次多项式，其朱利亚集$J$是$J$中的双可达点集的局部连通集，除了当$f(z)=z^2-2$是相应测度为1的切比雪夫二次多项式时，$J$是零。
Sierpiński curve Julia sets for quadratic rational maps
研究了在什么动力学条件下二次有理映射的朱利亚集是Sierpierski曲线
The measure of quadratic Julia-Lavaurs sets with a bounded type virtual Siegel disk is zero
我们在这里证明作者的论文的索赔之一。
The omega-limit sets of quadratic Julia sets
本文刻画了二次映射树朱利亚集的$\w$-极限集.我们使用鲍德温的符号表示这些空间作为一个非Hausdorff行程空间，并证明了二次映射与树突朱利亚集有阴影，也为所有这样的地图，一个封闭的不变集是一个点的$\w$-极限集当且仅当，它是内部链传递。
On biaccessible points in the Julia set of a Cremer quadratic polynomial
证明了Cremer二次多项式的朱利亚集中唯一可能的双可达点是Cremer不动点及其原像.这部分地回答了C. McMullen关于这样一个朱利亚集是否能包含任何双可达点的问题
Biaccessiblility in quadratic Julia sets II: The Siegel and Cremer cases
设$f$是一个二次多项式，它有一个无理中立不动点$α$ .设$z$是$f$的朱利亚集上的一个双可达点.然后：1.在西格尔的例子中，$z$的轨道最终一定会碰到$f$的临界点。2.在Cremer的例子中，$z$的轨道最终必须撞击不动点$α$。具有双可达临界点的Siegel多项式当然存在，但在Cremer情况下，双可达点可能永远不存在。作为推论，我们得出Siegel或Cremer二次多项式的朱利亚集中的双可达点集的Brolin测度为零的结论.
Julia Sets and wild Cantor sets
存在一致拟正则映射$f:\mathbb{R}^3 \to \mathbb{R}^3$，其朱利亚集是野生Cantor集.
Equilibrium measures on Julia sets of random quadratic polynomials
本文研究随机二次多项式复合序列的 Julia 集上的平衡测度，计算其 Hausdorff 维数并分析其与随机性尺度的关系。
For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n} (z) = z^2 + c_n$, which is the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disk. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia …
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
Julia Sets for Sequences of Monomials
研究黎曼球面上单项式非自治迭代产生的朱利亚集，通过系数和次数精确描述其相关朱利亚集，结合不变关系给出多种例子，并通过库拉托夫斯基极限阐述自治与非自治朱利亚集的联系。
Computability of Julia sets
本文解决了关于朱利亚集算法可计算性的大部分公开问题。特别地，我们提出了一个算法，用于构造二次的朱利亚集是不可计算的。我们还证明了多项式的满朱利亚集总是可计算的。
Laplacians on Julia Sets III: Cubic Julia Sets and Formal Matings
我们继续研究构造朱利亚集上的不变Laplacian，并研究其谱的性质。本文主要讨论两类例子：1）三次多项式$z^3 + c$的单临界点的朱利亚集; 2）二次朱利亚集的形式匹配.在本系列的前几篇文章中介绍的一般方案涉及将Julia集实现为具有标识的圆，并试图获得Laplacian作为从具有标识的圆模型导出的图上的图Laplacian的重正化极限。在三次朱利亚集的情况下，细节遵循为二次朱利亚集建立的模式，但对于配对，细节是相当具有挑战性的，我们只在一个例子中完全成功。一旦我们构造了拉普拉斯算子，我们就能够使用数值方法来近似本征值和本征函数。一个引人注目的观察数据是，三次朱利亚集的重数的所有特征空间（除了平凡的本征空间的常数）是偶数。对于前面研究过的二次朱利亚集，这样的情况是不成立的。我们能够解释这一点，是基于3是奇数这一事实，更确切地说，是因为二面角3对称群只有两个不同的一维不可约表示。
Rational maps whose Julia sets are Cantor circles
本文给出了一类其朱利亚集是Cantor圈的有理映射，并证明了每一个其Julia集是Cantor圈集的有理映射必在其对应的朱利亚集上与这类映射拓扑共轭.特别地，我们给出了某些有理映射的具体表达式，这些有理映射的朱利亚集是Cantor圈，但它们在其朱利亚集上不与任何McMullen映射拓扑共轭。此外，还构造了一类朱利亚集为Cantor圈的非双曲有理映射.
We study numerically the $α$- and $ω$-limits of the Newton maps of two of the most elementary families of polynomial transformations on the plane: those with a linear component and those with both components of degree two. Our results are fully consistent with some conjectures we posed in a recent work about the dynamics of Newton maps.
We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green functions and Parreau-Widom criterion for a special family of real generalized Julia sets.
Quasisymmetrically inequivalent hyperbolic Julia sets
我们给出了双曲有理映射的朱利亚集是同胚的但不是拟对称同胚的对的明确例子。
Lebesgue measure of Feigenbaum Julia sets
构造了朱利亚集具有正勒贝格测度的Feigenbaum二次多项式.他们提供了第一个例子的理性地图的双曲维数是不同的豪斯多夫维数的朱利亚集。相应的参数集具有正的豪斯多夫维数。
Polynomial skew products whose Julia sets have infinitely many symmetries
我们考虑C^2上多项式斜积的朱利亚集的对称性，它与旋转积是双有理共轭的。我们的主要结果给出了朱利亚集具有无穷多个对称性的多项式斜积的分类。
Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.
Quasisymmetries of Sierpiński carpet Julia sets
证明了若$ξ$是后临界有限有理映射的朱利亚集的Sierpienski地毯之间的拟对称同胚，则$ξ$是Möbius变换对朱利亚集的限制.这意味着一个后临界有限有理映射的Sierpienski地毯朱利亚集的拟对称同胚群是有限的。
On the Lebesgue measure of the Julia set of a quadratic polynomial
本文的目的是证明以下定理：设$p_a(z) = z^2+a$是一个二次多项式，它没有无理的中立周期点，并且不是无限可重整的.则朱利亚集的$J(p_a)$的勒贝格测度等于零。作为证明的一部分，我们讨论了临界点是{ \it持续常返}的一个性质，并将我们的结果与实一维映射的相应结果联系起来.特别是，我们表明，在持续经常性的情况下，限制$p_a|ω(0)$是拓扑最小的，并有零拓扑熵。Douady-Hubbard-Yoccoz刚性定理遵循这个结果。
On computability of Julia sets: answers to questions of Milnor and Shub
在这篇笔记中，我们给出了J.Milnor和M.Shub向我们提出的问题的答案，这些问题进一步阐明了不可计算的朱利亚集的结构。
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。
Hausdorff measure of Julia sets in the exponential family
本文研究了指数映射的朱利亚集和逃逸集关于某些度规函数的Hausdorff测度。我们给出了度规函数的增长条件，这些条件分别意味着测度为零或无穷大。
Recurrent and periodic points in dendritic Julia sets
我们涉及到树枝状朱利亚集中的周期点和回归点。这推广了区间动力学的著名结果。
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。
The Inverse Iteration Method for Julia Sets in the 3-Dimensional Space
在这篇文章中，我们介绍了自适应逆迭代法来产生双复朱利亚集相关的多项式映射$w^2+c$。这个结果是基于双复朱利亚集作为一个特殊的双复Carnival集的边界的充分刻画和对$w^2+c$不动点的研究.逆迭代法特别用于在通常的三维空间中生成和显示作为枝晶的双复朱利亚集。
For any polynomial diffeomorphism $f$ of ${\Bbb C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is semi-analytic.
Porosity of Collet-Eckmann Julia sets
证明了Riemann球面上满足Collet-Eckmann条件且不含抛物周期点的有理映射的朱利亚集是平均多孔的，如果它不是整个球面.因此，朱利亚集的Minkowski维数小于2。
The Fatou-Julia decomposition is significant in the study of iterations of holomorphic mappings. Such a decomposition can be also considered for foliations in a unified manner. Although the decomposition will be fundamental in the study, it is not easy to decide the decomposition. In this article, we give a sufficient condition for open sets to be contained in Fatou sets. We also discuss relations between Fatou--Julia decompositions and minimal sets.
Shapes of Polynomial Julia Sets
复平面上的任何Jordan曲线都可以在Hausdorff拓扑中用多项式的朱利亚集任意好地逼近。不相交的Jordan域的有限集合可以用有理映射的吸引盆来近似。
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…
No smooth Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with positive entropy
对$\mathbb{C}^2$的任意正熵多项式同态$f$，其Julia集及其逆$f^{-1}$的Julia集都不是$C^1$光滑的有边界流形.
Buried Points in Julia Sets
我们给一个介绍埋点在朱利亚集和一个列表的问题埋点，写鼓励爱好者的拓扑结构和动力学工作在这些问题上。
Hausdorff dimension and biaccessibility for polynomial Julia sets
研究了任意次连通多项式朱利亚集的双可达点集$d\geq 2$ .证明了除非朱利亚集是区间，否则双可达点对应的外角集的豪斯多夫维数小于1.这加强了斯坦尼斯拉夫·斯米尔诺夫和安娜·兹杜尼克的定理：他们证明了同一组外角的一维测度为零。
Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form $x+τy$ for $x,y \in \mathbb{R}$, and $τ^2 = 1$ but $τ\neq \pm 1$, are the natural number system in which to encode geometric properties of the Minkowski space $\mathbb{R}^{1,1}$. We show that the hyperbolic analog of the Mandelbrot set parameterizes connectedness of hyperbolic Julia sets. We give a wall-and-chamber decom…
On the Hausdorff dimension of Julia sets of some real polynomials
证明了多项式$z\mapsto z^d+c$（$d$是偶自然数）的朱利亚集的豪斯多夫维数的$c$实数的上确界大于$2d/(d+1)$ .
For any integers $d\ge 3$ and $n\ge 1$, we construct a hyperbolic rational map of degree $d$ such that it has $n$ cycles of the connected components of its Julia set except single points and Jordan curves.
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.
We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope some of them occur useful to achieve a fast approximation from below of the hyperbolic Hausdorff dimension of Julia set.
Non-persistently recurrent points, qc-surgery and instability of rational maps with totally disconnected Julia sets
设$ R $是具有全不连通的朱利亚集$ J(R). $的有理映射.若$ J(R) $上的后临界集包含一个非持久递归（或锥）点，则证明了该映射$ R $不可能是结构稳定映射.
In this paper we introduce the notion of dynamical systems over the class of the normed real nonassociative algebras not necessarily finite-dimensional, generalize the classical filled Julia and Mandelbrot sets over the complex numbers, investigate their orbits and make a topological analysis. Applications of the obtained results are made.
Let $d(δ)$ denote the Hausdorff dimension of the Julia set of the polynomial $f_δ(z)=z^2-2+δ$. In this paper we will study the directional derivative of the function $d$ along directions landing at the parameter $0$, which corresponds to $-2$ in the case of family $p_c(z)=z^2+c$. We will consider all directions, except the one $δ\in\mathbb{R}^+$, which is inside the Mandelbrot set. We will prove asymptotic formula for the directional derivative of $d$. Moreover, we will see that the derivative is negative for all directions in the closed left half-plane. Computer calculations show that it is n…
Tricomplex Distance Estimation for Filled-in Julia Sets and Multibrot Sets
本文提出一种距离估计公式，用于光线追踪由形如 $η^p+c$ 的三复数多项式生成的填充 Julia 集与 Multibrot 集的三维切片。
In holomorphic semigroup dynamics, Julia set is in general backward invariant and so some fundamental results of classical complex dynamics can not be generalized to semigroup dynamics. In this paper, we define completely invariant Julia set of transcendental semigroup and we see how far the results of classical transcendental dynamics generalized to transcendental semigroup dynamics.
The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Chéritat. Since then many efforts have been devoted to finding out new classes of rational maps with nowhere dense Julia sets having positive area. So far, all known examples of this kind are renormalizable with only one exception which is a quadratic polynomial. In this paper, by developing a new approach, we prove that there exists a non-renormalizable cubic polynomial having a Julia set with positive area.
A tree, embedded into plane, is a dessin d'enfant and its Belyi function is a polynomial --- Shabat polynomial. The Zapponi form of this polynomial is unique, so we can correspond to an embedded tree the Julia set of its Shabat-Zapponi polynomial. In this purely experimental work we study relations between the form of a tree and properties (form, connectedness, Hausdorff dimension) of its Julia set.
Irreducible Julia sets of rational functions
证明了多项式朱利亚集是一个不可约连续统，它要么是一个弧，要么是一个不可约连续统.对于更一般的有理函数情形，当朱利亚集是不可约连续统且所有不可分解子连续统内部为空时，给出了动力学的拓扑模型.
A. Sannami constructed an example of the differentiable Cantor set embedded in the real line whose difference set has a positive measure. In this paper, we generalize the definition of the difference sets for sets of the two dimensional Euclidean space as the sets of vectors between two sets, and estimate their measures. For the quadratic map Q_c(z)=z^2+c, we obtain that the measure of the difference set of its Julia set vanishes if |c|>3+sqrt{3}.
Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts
证明了对于具有对数道的亚纯映射（例如$\mathcal B$类中具有有限个极点的整映射或亚纯映射），朱利亚集包含一个大于1的豪斯多夫维数的紧不变双曲康托尔集.因此，朱利亚集的双曲维数大于1。
The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
证明了Mandelbrot集$M$的边界具有豪斯多夫维数2，并且对于一般的$c \in \bM$，$z \mapsto z^2+c$的朱利亚集也具有豪斯多夫维数2.证明是基于对抛物型周期点分支的研究。
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 …
Non-Autonomous Julia Sets with Invariant Sequences of Measurable Line Fields
非不变线场猜想是传统复动力学中的主要突出问题之一。在本文中，我们考虑非自治的迭代，其中一个工程与多项式序列的成分与适当的范围的程度和系数。我们表明，自然推广的没有不变的线场猜想，这个设置是不正确的。特别地，我们构造了一个二次多项式序列，它的迭代朱利亚集都有正面积，并且它有一个不变的可测线场序列，其支撑是这些迭代朱利亚集.
A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set M, the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on $c$ and find explicit bounds at the tip of M for most real parameters in the the sense of 1-dimensional Lebesgue measure.
Uniformly quasiregular maps of $\mathbb{R}^n$ with Cantor Julia sets
研究\(n\geq3\)时\(\mathbb{R}^n\)的映射，构造多项式型一致拟正则自映射，使朱利亚集为康托集。
Proof of the Branner-Hubbard conjucture on Cantor Julia sets
利用Kozlovski，Shen和van Strien构造的临界片嵌套序列，以及Kahn和Lyubich最近证明的覆盖引理，证明了多项式的朱利亚集是康托尔集当且仅当包含临界点的填充朱利亚集的每个分支是非周期的.这个结果是Branner和Hubbard在1992年提出的一个猜想。
On Fibers and Renormalization of Julia Sets and Multibrot Sets
我们继续在IMS预印本1998/12和1998/13 a中开始的关于纤维的Mandelbrot集和Multibrot集以及朱利亚集的描述。这些集合的局部连通性问题在纤维方面进行了讨论，并成为纤维的平凡性问题。在本文中，重点是在重整化和其他手术程序下的纤维的行为。我们证明了Mandelbrot集和Multibrot集的纤维的平凡性在调谐映射和其他（部分）同胚下保持不变。同样，我们证明了单临界多项式的平凡纤维的朱利亚集下的重整化和其他手术程序，如Branner-Douady同胚。我们的结论与各种应用程序的二次多项式和它的参数空间：我们确定嵌入路径内的Mandelbrot集，我们表明，彼得森定理二次朱利亚集与西格尔磁盘的有界型推广从周期1到任意周期，使他们都有平凡的纤维，从而局部连接。
We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。