Filled Julia sets with empty interior are computable

Non-computable impressions of computable external rays of quadratic polynomials

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.

Computability of the Julia set. Nonrecurrent critical orbits

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.

On computability of Julia sets: answers to questions of Milnor and Shub

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.

Computability of Julia sets

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.

单项式序列的朱利亚集

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。

Real quadratic Julia sets can have arbitrarily high complexity

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.

Fatou Set, Julia Set and Escaping Set in Holomorphic Subsemigroup Dynamics

We investigate to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of transcendental semigroup and on the basis of this set, we prove that Fatou set and escaping set of transcendental semigroup $ S $ are non-empty.

Irreducible Julia sets of rational functions

We prove that a polynomial Julia set which is a finitely irreducible continuum is either an arc or an indecomposable continuum. For the more general case of rational functions, we give a topological model for the dynamics when the Julia set is an irreducible continuum and all indecomposable subcontinua have empty interior.

具有空有理层压的三次复多项式的立即重整化

Immediate renormalization of cubic complex polynomials with empty rational lamination

本文研究三次复多项式的立即重整化性质，证明在Julia集无（预）周期割点的条件下，不属于不变填充Julia集的唯一临界点必为回归点。

Quadratic Dynamics Over Hyperbolic Numbers

Hyperbolic numbers are a variation of complex numbers, but their dynamics is quite different. The hyperbolic Mandelbrot set for quadratic functions over hyperbolic numbers is simply a filled square, and the filled Julia set for hyperbolic parameters inside the hyperbolic Mandelbrot set is a filled rectangle. For hyperbolic parameters outside the hyperbolic Mandelbrot set, the filled Julia set has 3 possible topological descriptions, if it is not empty, in contrast to the complex case where it is always a non-empty totally disconnected set. These results were proved in [1,2,4,5,6,7] and are rev…

Julia sets of random exponential maps

For a sequence $(λ_n)$ of positive real numbers we consider the exponential functions $f_{λ_n} (z) = λ_n e^z$ and the compositions $F_n = f_{λ_n} \circ f_{λ_{n-1}} \circ ... \circ f_{λ_1}$. For such a non-autonomous family we can define the Fatou and Julia sets analogously to the usual case of autonomous iteration. The aim of this document is to study how the Julia set depends on the sequence $(λ_n)$. Among other results, we prove the Julia set for a random sequence $\{λ_n \}$, chosen uniformly from a neighbourhood of $\frac{1}{e}$, is the whole plane with probability $1$. We also prove the Ju…

一种动态算法用于多项式时间内计算双曲Julia集

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。

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.

亚纯函数的有界 Fatou 分支与 Julia 分支

Bounded Fatou and Julia components of meromorphic functions

本文完整刻画亚纯函数中有界 Fatou 分支与 Julia 分支的拓扑特征，并借助逼近理论构造含游荡连续统的亚纯函数给出证明。

具有非空Fatou集的慢回复Collet-Eckmann映射

Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

该研究针对Julia集非全球面、临界点慢回复的有理Collet-Eckmann映射，证明其在固定临界点阶数的族中是双曲映射的Lebesgue密度点，单临界点情形下在任意次数≥2的有理映射全空间中也成立。

A characterization of Sierpinski carpet Rational maps

In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet

Bifurcations in the elementary Desboves family

We give an example of a family of endomorphisms of $\mathbb P^2 (\mathbb C)$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non empty interior.

The dynamics of quasiregular maps of punctured space

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties…

复Hénon映射的Julia集的可计算性：吸引与中性周期轨道的作用

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。

三次有理映射的斥周期点的密度

Density of repelling periodic points for cubic rational maps

针对三次有理映射，证明了其在$\mathbf C_p$上的斥周期点密度猜想，通过反证法结合系数分析，确定了特征为2和3时的情况，得出经典Julia集为空的结论。

Non-Archimedean Hénon maps, attractors, and horseshoes

We study the dynamics of the Hénon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Hénon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Hénon maps over ${\mathbb Q}_3$, this attractor is infinite and supports an SRB…

Computability of Brolin-Lyubich Measure

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…

与复动力系统相关的C*-代数的Cartan子代数

Cartan subalgebras of C*-algebras associated with complex dynamical systems

本文研究复动力系统中C*-代数的结构，针对有理函数在Julia集、Fatou集或黎曼球面上的作用，给出$C_0(X)$成为$\mathcal{O}_R(X)$的Cartan子代数的等价条件，推广了Kajiwara和Watatani关于极大交换子代数的结果。

关于复平面上超越位移型映射的两个注记

Two remarks on transcendental shift-like maps on $\mathbb{C}^N$

本文研究了超越位移型映射的Julia集非空性，并给出了具有逃逸游荡域的例子，对比了多项式位移映射的动力学行为。

特征簇自同构群的动力学与 Painlevé 6 的 Fatou/Julia 分解

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。

Higher bifurcations for polynomial skew-products

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate \emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an …

复动力系统中轨道集的维数

Dimensions of orbital sets in complex dynamics

该研究针对黎曼球面上的紧子集与次数不小于2的有理映射，分析其反向轨道集的上盒维数与Julia集、初始集上盒维数的关系，将非齐次迭代函数系的相关研究拓展至复动力系统领域。

$\boldsymbol{\text{ℂ}^2}$的Axiom A多项式斜积的可计算性

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。