## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|CS2401

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂系统与重整化Complex Systems and Reengineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂系统与重整化Complex Systems and Reengineering代写方面经验极为丰富，各种代写复杂系统与重整化Complex Systems and Reengineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|The hyperbolic metric

A Riemann surface is hyperbolic if its universal cover is isomorphic to the upper halfplane $\mathbb{H}$. The hyperbolic metric or Poincaré metric on such a Riemann surface is the unique complete conformal metric of constant curvature $-1$.
By the Schwarz lemma [Ah2, $\S 1-2]$ one knows:
Theorem 2.2 A holomorphic map $f: X \rightarrow Y$ between hyperbolic Riemann surfaces does not increase the Poincaré metric, and $f$ is a local isometry if and only if $f$ is a covering map.

The Poincaré metric is defined on any region $U \subset \widehat{\mathbb{C}}$ provided $|\widehat{\mathbb{C}}-U|>2$. If $U$ is not connected, we define its Poincaré metric component by component.
The hyperbolic metric on the upper halfplane $\mathbb{H}$ is given by:
$$\rho=\frac{|d z|}{\operatorname{Im}(z)}$$
on the unit disk $\Delta$, by:
$$\rho=\frac{2|d z|^2}{1-|z|^2}$$
on the punctured disk $\Delta^$, by: $$\rho=\frac{|d z|}{|z| \log (1 /|z|)}$$ and on the annulus $A(R)$ by: $$\rho=\frac{\pi / \log R}{\sin (\pi \log |z| / \log R)} \frac{|d z|}{|z|} .$$ The last two formulas can be verified using the covering maps $z \mapsto$ $\exp (i z)$ from $\mathbb{H}$ to $\Delta^$ and $z \mapsto z^{\log R / \pi i}$ from $\mathbb{H}$ to $A(R)$

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Metric aspects of annuli

Let $V$ be a Riemann surface which is topologically a disk, and let $E \subset V$ have compact closure. It is convenient to have a measurement of the amount of space around $E$ in $V$. For this purpose we define $\bmod (E, V)=\sup {\bmod (A): A \subset V$ is an annulus enclosing $E}$.
(This means $E$ should lie in the compact component of $V-A$.) Note that $\bmod (E, V)=\infty$ if $V$ is isomorphic to $\mathbb{C}$ or if $E$ is a single point.
Now suppose $V$ is hyperbolic, and let $\operatorname{diam}(E)$ denote diameter of $E$ in the hyperbolic metric on $V$.

Theorem 2.4 The hyperbolic diameter and modulus of $E$ are inversely related:
$$\operatorname{diam}(E) \rightarrow 0 \Longleftrightarrow \bmod (E, V) \rightarrow \infty$$
and
$$\operatorname{diam}(E) \rightarrow \infty \Longleftrightarrow \bmod (E, V) \rightarrow 0 .$$
More precisely,
$$\operatorname{diam}(E) \asymp \exp (-2 \pi \bmod (E, V))$$
when either side is small, while
$$\frac{C_1}{\operatorname{diam}(E)} \geq \bmod (E, V) \geq C_2 \exp (-\operatorname{diam}(E))$$
when the diameter is large.

Proof. The first estimate follows from existence of a round annulus as guaranteed by Theorem 2.1. The second follows using estimates for the Grötzsch modulus [LV, §II.2].

The relation of modulus to hyperbolic diameter is necessarily imprecise when the diameter is large. For example, for $r<1$ the sets $E_1=[-r, r]$ and $E_2=\Delta(r)$ have the same hyperbolic diameter $d$ in the unit disk, but for $r$ near $1, \bmod \left(E_1, \Delta\right) \asymp 1 / d$ while $\bmod \left(E_2, \Delta\right) \asymp e^{-d}$

The next result controls the Euclidean geometry of an annulus of definite modulus.

# 复杂系统与重整化代写

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|The hyperbolic metric

Poincaré 度量定义在任何区域 $U \subset \widehat{\mathbb{C}}$ 假如 $|\widehat{\mathbb{C}}-U|>2$. 如果 $U$ 不连通，我们逐个定义它的庞加莱度量 分量。

$$\rho=\frac{|d z|}{\operatorname{Im}(z)}$$

$$\rho=\frac{2|d z|^2}{1-|z|^2}$$

$$\rho=\frac{|d z|}{|z| \log (1 /|z|)}$$

$$\rho=\frac{\pi / \log R}{\sin (\pi \log |z| / \log R)} \frac{|d z|}{|z|} .$$

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Metric aspects of annuli

$$\operatorname{diam}(E) \rightarrow 0 \Longleftrightarrow \bmod (E, V) \rightarrow \infty$$

$$\operatorname{diam}(E) \rightarrow \infty \Longleftrightarrow \bmod (E, V) \rightarrow 0$$

$$\operatorname{diam}(E) \asymp \exp (-2 \pi \bmod (E, V))$$

$$\frac{C_1}{\operatorname{diam}(E)} \geq \bmod (E, V) \geq C_2 \exp (-\operatorname{diam}(E))$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|SE749

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂系统与重整化Complex Systems and Reengineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂系统与重整化Complex Systems and Reengineering代写方面经验极为丰富，各种代写复杂系统与重整化Complex Systems and Reengineering相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Summary of contents

We begin in $\S 2$ with a resume of results from hyperbolic geometry, geometric function theory and measure theory. Then we introduce the theory of iterated rational maps, and study their measurable dynamics in $\S 3$.
Here one may see the first instance of a general philosophy:
Expanding dynamics promotes a measurable line field to a holomorphic line field.
This philosophy has precursors in [Sul1] and classical arguments in ergodic theory.

In $\S 4$ we discuss holomorphic motions and structural stability in general families of rational maps. Then we specialize to the Mandelbrot set, and explain the equivalence of Conjectures $1.2$ and 1.5.
In $\S 5$, we develop compactness results to apply the expansion philosophy in the context of renormalization. We also introduce the polynomial-like maps of Douady and Hubbard, which play a fundamental role in renormalization.

In $\S 6$, we turn to polynomials and describe the use of external rays in the study of their combinatorics.

With this background in place, the theory of renormalization is developed in $\S$ 7. New types of renormalization, unrelated to “tuning”, were discovered in the course of this development; examples are presented in $\S 7.4$.
$\S 8$ describes infinitely renormalizable quadratic polynomials. Included is an exposition of the Yoccoz puzzle, a Markov partition for the dynamics of a quadratic polynomial. Theorem $1.6$ is discussed along with work of Lyubich and Shishikura.

In $\S 9$ we define robust quadratic polynomials, and prove their postcritical sets have measure zero. This assertion is essential for applying the expansion philosophy, because we only obtain expansion in the complement of the postcritical set.

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|The modulus of an annulus

Any Riemann surface $A$ with $\pi_1(A) \cong \mathbb{Z}$ is isomorphic to $\mathbb{C}^, \Delta^$ or the standard annulus $A(R)$ for some $R \in(1, \infty)$. In case $A$ is isomorphic to $A(R)$, the modulus of $A$ is defined by
$$\bmod (A)=\frac{\log R}{2 \pi} .$$
Thus $A$ is conformally isomorphic to a right cylinder of circumference one and height $\bmod (A)$. By convention $\bmod (A)=\infty$ in the other two cases.

An annulus $B \subset \mathbb{C}$ is round if it is bounded by concentric Euclidean circles (so $B$ has the form ${z: r<|z-c|<s}$ ).

Theorem 2.1 (Round annulus) Any annulus $A \subset \mathbb{C}$ of sufficiently large modulus contains an essential round annulus $B$ with $\bmod (A)=$ $\bmod (B)+O(1)$

Here essential means $\pi_1(B)$ injects into $\pi_1(A)$, i.e. $B$ separates the boundary components of $A$.

Proof. We may assume $\widehat{\mathbb{C}}-A$ consists of two components $C$ and $D$, where $0 \in C$ and $\infty \in D$. Let $z_1 \in C$ maximize $|z|$ over $C$, and let $z_2 \in D$ minimize $|z|$ over $D$. By Teichmüller’s module theorem [LV, §II.1.3]
$$\bmod (A) \leq \frac{1}{\pi} \mu\left(\sqrt{\frac{\left|z_1\right|}{\left|z_1\right|+\left|z_2\right|}}\right)$$
where $\mu(r)$ is a positive decreasing function of $r .{ }^1$ Thus $\left|z_1\right|<\left|z_2\right|$ if $\bmod (A)$ is sufficiently large, in which case $A$ contains a round annulus $B=\left{z:\left|z_1\right|<|z|<\left|z_2\right|\right}$. Moreover, once $\left|z_1\right|<\left|z_2\right|$ we have
$$\bmod (A) \leq \frac{1}{\pi} \mu\left(\sqrt{\frac{\left|z_1\right|}{2\left|z_2\right|}}\right) \leq \bmod (B)+\frac{5 \log 2}{2 \pi}$$
by the inequality $\mu(r)<\log (4 / r)$ [LV, eq. (2.10) in §II.2.3].
An alternative proof can be based on the following fact: any sequence of univalent maps $f_n:\left{z: 1 / R_n<|z|<R_n\right} \rightarrow \mathbb{C}^*$, with $f_n(1)=1$ and with the image of $f$ separating 0 from $\infty$, converges to the identity as $R_n \rightarrow \infty$.

# 复杂系统与重整化代写

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Summary of contents

§§8描述可无限重归一化的二次多项式。包括对 Yoccoz 难题的阐述，这是二次多项式动力学的马尔可夫分区。定理1.6与 Lyubich 和 Shishikura 的作品一起讨论。

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|The modulus of an annulus

\bmod (A) \leq \frac{1}{\pi} \mu\left(\sqrt{\frac{\left|z_1\right|}{\left|z_1\right|+\left|z_2\right|}}\right)
$$在哪里 \mu(r) 是正减函数 r .{ }^1 因此 \left|z_1\right|<\left|z_2\right| 如果 \bmod (A) 足够大，在这种情况下 A 包含一个圆环 \mathrm{B}=\backslash left \left{z: \backslash\right. left \mid z_{-} 1 \backslash right |<| z \mid<\backslash left \mid z_{-}_\right } | \text { right } } \text { . 而且，曾经 } | z _ { 1 } | < | z _ { 2 } | \text { 我们有 }$$
\bmod (A) \leq \frac{1}{\pi} \mu\left(\sqrt{\frac{\left|z_1\right|}{2\left|z_2\right|}}\right) \leq \bmod (B)+\frac{5 \log 2}{2 \pi}
$$由不平等 \mu(r)<\log (4 / r) [LV，当量。(2.10) 在§II.2.3]。 另一种证明可以基于以下事实: 任何单价映射序列 \mathrm{f}{-} n: \backslash left \left{z: 1 / R{-} n<|z|<R_{-} n \backslash r i g h t\right} \backslash r i g h t a r r o w \backslash m a t h b b{C}^{\wedge *} ， 和 f_n(1)=1 和形象 f 将 0 与 \infty ，收敛到身 份为 R_n \rightarrow \infty. 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## MATLAB代写 MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 ## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|COM3523 如果你也在 怎样代写复杂系统与重整化Complex Systems and Reengineering这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 随机图是一种图，其中图顶、图边和它们之间的连接数等属性是以某种随机方式确定的。 statistics-lab™ 为您的留学生涯保驾护航 在代写复杂系统与重整化Complex Systems and Reengineering方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复杂系统与重整化Complex Systems and Reengineering代写方面经验极为丰富，各种代写复杂系统与重整化Complex Systems and Reengineering相关的作业也就用不着说。 我们提供的复杂系统与重整化Complex Systems and Reengineering及其相关学科的代写，服务范围广, 其中包括但不限于: • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等概率论 • Advanced Mathematical Statistics 高等数理统计学 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 ## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Complex dynamics This work presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Around 1920 Fatou and Julia initiated the theory of iterated rational maps$$
f: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}

on the Riemann sphere. More recently methods of geometric function theory, quasiconformal mappings and hyperbolic geometry have contributed to the depth and scope of research in the field. The intricate structure of the family of quadratic polynomials was revealed by work of Douady and Hubbard [DH1], [Dou1]; analogies between rational maps and Kleinian groups surfaced with Sullivan’s proof of the no wandering domains theorem [Sul3] and continue to inform both subjects [Mc2].

It can be a subtle problem to understand a high iterate of a rational map $f$ of degree $d>1$. There is tension between expanding features of $f$-such as the fact that its degree tends to infinity under iteration – and contracting features, such as the presence of critical points. The best understood maps are those for which the critical points tend to attracting cycles. For such a map, the tension is resolved by the concentration of expansion in the Julia set or chaotic locus of the map, and the presence of contraction on the rest of the sphere.

The central goal of this work is to understand a high iterate of a quadratic polynomial. The special case we consider is that of an infinitely renormalizable polynomial $f(z)=z^2+c$.

For such a polynomial, the expanding and contracting properties lie in a delicate balance; for example, the critical point $z=0$ belongs to the Julia set and its forward orbit is recurrent. Moreover high iterates of $f$ can be renormalized or rescaled to yield new dynamical systems of the same general shape as the original map $f$.

This repetition of form at infinitely many scales provides the basic framework for our study. Under additional geometric hypotheses, we will show that the renormalized dynamical systems range in a compact family. Compactness is established by combining universal estimates for the hyperbolic geometry of surfaces with distortion theorems for holomorphic maps.

With this information in hand, we establish quasiconformal rigidity of the original polynomial $f$. Rigidity of $f$ supports conjectures about the behavior of a generic complex dynamical system, as described in the next section.

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Central conjectures

We now summarize the main problems which motivate our work.
Let $f: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ be a rational map of the Riemann sphere to itself of degree $d>1$. The map $f$ is hyperbolic if its critical points tend to attracting periodic cycles under iteration. Within all rational maps, the hyperbolic ones are among the best behaved; for example, when $f$ is hyperbolic there is a finite set $A \subset \widehat{\mathbb{C}}$ which attracts all points in an open, full-measure subset of the sphere (see $\S 3.4$ ).

One of the central problems in conformal dynamics is the following:

Conjecture $1.1$ (Density of hyperbolicity) The set of hyperbolic rational maps is open and dense in the space Rat $_d$ of all rational maps of degree $d$.

Openness of hyperbolic maps is known, but density is not. In some form this conjecture goes back to Fatou (see $\S 4.1$ ).

Much study has been devoted to special families of rational maps, particularly quadratic polynomials. Every quadratic polynomial $f$ is conjugate to one of the form $f_c(z)=z^2+c$ for a unique $c \in \mathbb{C}$. Even this simple family of rational maps exhibits a full spectrum of dynamical behavior, reflecting many of the difficulties of the general case. Still unresolved is:

Conjecture 1.2 The set of $c$ for which $z^2+c$ is hyperbolic forms ип ореп dense subsel of lhe complex plane.

The Mandelbrot set $M$ is the set of $c$ such that under iteration, $f_c^n(0)$ does not tend to infinity; here $z=0$ is the unique critical point of $f_c$ in $\mathbb{C}$. A component $U$ of the interior of $M$ is hyperbolic if $f_c$ is hyperbolic for some $c$ in $U$. It is known that the maps $f_c$ enjoy a type of structural stability as $c$ varies in any component of $\mathbb{C}-\partial M$; in particular, if $U$ is hyperbolic, $f_c$ is hyperbolic for every $c$ in $U$ (see $\S 4)$. It is clear that $f_c$ is hyperbolic when $c$ is not in $M$, because the critical point tends to the superattracting fixed point at infinity. Thus an equivalent formulation of Conjecture $1.2$ is:

Conjecture 1.3 Every component of the interior of the Mandelbrot set is hyperbolic.

An approach to these conjectures is developed in [MSS] and [McS], using quasiconformal mappings. This approach has the advantage of shifting the focus from a family of maps to the dynamics of a single map, and leads to the following:

Conjecture $1.4$ (No invariant line fields) A rational map $f$ carries no invariant line field on its Julia set, except when $f$ is double covered by an integral torus endomorphism.

# 复杂系统与重整化代写

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Complex dynamics

1920 年前后，Fatou 和 Julia 提出了迭代有理映射理论

F:C^→C^

## 数学代写|复杂系统与重整化代写Complex Systems and Reengineering代考|Central conjectures

[MSS] 和 [McS] 中使用拟共形映射开发了这些猜想的方法。这种方法的优点是将焦点从一系列地图转移到单个地图的动态，并导致以下结果：

## 有限元方法代写

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