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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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}


$\mathrm{f}{-} n: \backslash$ 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 $R_n \rightarrow \infty$.

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## MATLAB代写

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