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## 数学代写|遍历理论代考Ergodic theory代考|Asymptotic h-Expansiveness

This Chapter is devoted to the investigation of the weak expansion properties of expanding Thurston maps. The main theorem for this chapter is the following.
Theorem 6.1 Let $f: S^2 \rightarrow S^2$ be an expanding Thurston map. Then $f$ is asymptotically $h$-expansive if and only if $f$ has no periodic critical points. Moreover, $f$ is not h-expansive.

As an immediate consequence of this theorem and J. Buzzi’s result on the asymptotic $h$-expansiveness of $C^{\infty}$-maps on compact Riemannian manifolds [Buz97], we get the following corollary, which partially answers a question of $\mathrm{K}$. Pilgrim (see Problem 2 in [BM10, Sect. 21]).

Corollary 6.2 An expanding Thurston map with at least one periodic critical point cannot be conjugate to a $C^{\infty}$-map from the Euclidean 2-sphere to itself.

Remark $6.3$ Corollary $6.2$ can also be proved using an elementary argument which we include here. Suppose that a $C^{\infty}$-map $f: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ is an expanding Thurston map with a periodic critical point $p$. We can assume that $p=0$. By Theorem 2.16, there exists $N \in \mathbb{N}$ such that the $C^{\infty}$-map $F=f^N$ is an expanding Thurston map with a fixed critical point 0 such that there exists a Jordan curve $\mathscr{C} \subseteq S^2$ with $F(\mathscr{C}) \subseteq \mathscr{C}$ and post $F \subseteq \mathscr{C}$. Then there exists $r>0$ such that the Jacobian determinant $\operatorname{det} D F$ satisfies $|\operatorname{det} D F(z)|<\frac{1}{2}$ for all $z \in B_\rho(0, r)$. Here $\rho$ is the Euclidean metric on $\mathbb{C}$. Since $F$ is an expanding Thurston map, there exists $n \in \mathbb{N}$ such that the $n$-flower $W^n(0) \in \mathbf{W}^n(F, \mathscr{C})$ is a subset of $B_\rho(0, r)$. Note that $F\left(W^n(0)\right)=W^{n-1}(0)$. Thus $m\left(W^{n-1}(0)\right) \leq \int_{W^n(0)}|\operatorname{det} D F| \mathrm{d} m \leq \frac{1}{2} m\left(W^n(0)\right)$, where $m$ is the Lebesgue measure on $\mathbb{C}$. On the other hand, it is clear that $W^n(0) \subsetneq W^{n-1}(0)$ since $F$ is an expanding Thurston map. Since flowers are open, we get that $m\left(W^n(0)\right)<m\left(W^{m-1}(0)\right)$, a contradiction.

## 数学代写|遍历理论代考Ergodic theory代考|Some Properties of Expanding Thurstons Maps

We need the following three lemmas for the proof of the asymptotic $h$-expansiveness of expanding Thurston maps with no periodic critical points.

Lemma 6.6 (Uniform local injectivity away from the critical points) Let $f, d$ satisfy the Assumptions. Then there exist a number $\delta_0 \in(0,1]$ and a function $\tau:\left(0, \delta_0\right] \rightarrow$ $(0,+\infty)$ with the following properties:
(i) $\lim {\delta \rightarrow 0} \tau(\delta)=0$. (ii) For each $\delta \leq \delta_0$, the map $f$ restricted to any open ball of radius $\delta$ centered outside the $\tau(\delta)$-neighborhood of crit $f$ is injective, i.e., $\left.f\right|{B_d(x, \delta)}$ is injective for each $x \in S^2 \backslash N_d^{\tau(\delta)}($ crit $f)$.

This lemma is straightforward to verify, but for the sake of completeness, we include the proof here.

Proof We first define a function $r: S^2 \backslash$ crit $f \rightarrow(0,+\infty)$ in the following way
$$r(x)=\sup \left{R>0|f|_{B_d(x, R)} \text { is injective }\right},$$
for $x \in S^2 \backslash$ crit $f$. Note that $r(x) \leq d(x$, crit $f)<+\infty$ for each $x \in S^2 \backslash$ crit $f$. We also observe that the supremum is attained, since otherwise, suppose $f(y)=f(z)$ for some $y, z \in B_d(x, r(x))$, then $f$ is not injective on the ball $B_d\left(x, R_0\right)$ containing $y$ and $z$ with $R_0=\frac{1}{2}(r(x)+\max {d(x, y), d(x, z)})<r(x)$, a contradiction.
We claim that $r$ is continuous.

## 数学代写|遍历理论代考Ergodic theory代考|Asymptotic h-Expansiveness

$m\left(W^{n-1}(0)\right) \leq \int_{W^{n(0)}}|\operatorname{det} D F| \mathrm{d} m \leq \frac{1}{2} m\left(W^n(0)\right)$ ， 在哪里 $m$ 是勒贝格则度 $\mathbb{C}$. 另一方面，很明显 $W^n(0) \subsetneq W^{n-1}(0)$ 自从 $F$ 是一个扩展的瑟斯顿地图。既然花开了，我们就明白了 $m\left(W^n(0)\right)<m\left(W^{m-1}(0)\right)$ ，矛盾。

## 数学代写|遍历理论代考Ergodic theory代考|Some Properties of Expanding Thurstons Maps

(i) $\$ \backslash \lim {\mid$delta$\mid$rightarrow 0$} \mid$tau$(\mid$delta)$=0$. (ii)Foreach$\mid$delta$\mid$leq$\mid$delta_ 0 , themapF restrictedtoanyopenballofradius$\mid$三角洲centeredoutsidethe$\mid$数字 ($\mid$三角洲) -neighborhoodofcritF 这个引理很容易验证，但为了完整起见，我们在这里包含证明。 证明我们首先定义一个函数$r: S^2 \backslash$展击$f \rightarrow(0,+\infty)$通过以下方式 〈left 缺少或无法识别的分隔符 为了$x \in S^2 \backslash$暴击$f$. 注意$r(x) \leq d(x$，暴击$f)<+\infty$每个$x \in S^2 \backslash$暴击$f$. 我们还观㟯到达到了上界，否则，假设$f(y)=f(z)$对于一些$y, z \in B_d(x, r(x))$，然后$f$球上没有内射$B_d\left(x, R_0\right)$含有$y$和$z$和$R_0=\frac{1}{2}(r(x)+\max d(x, y), d(x, z))<r(x)$， 矛盾。 我们声称$r\$ 是连绖的。

## 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。