Posted on Categories:Ergodic theory, 数学代写, 遍历理论

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|遍历理论代考Ergodic theory代考|Covers and Partitions

Let $(X, d)$ be a compact metric space and $g: X \rightarrow X$ a continuous map.
A cover of $X$ is a collection $\xi=\left{A_j \mid j \in J\right}$ of subsets of $X$ with the property that $\bigcup \xi=X$, where $J$ is an index set. The cover $\xi$ is an open cover if $A_j$ is an open set for each $j \in J$. The cover $\xi$ is finite if the index set $J$ is a finite set.

A measurable partition $\xi$ of $X$ is a cover $\xi=\left{A_j \mid j \in J\right}$ of $X$ consisting of countably many mutually disjoint Borel sets $A_j, j \in J$, where $J$ is a countable index set. For $x \in X$, we denote by $\xi(x)$ the unique element of $\xi$ that contains $x$.

Let $\xi=\left{A_j \mid j \in J\right}$ and $\eta=\left{B_k \mid k \in K\right}$ be two covers of $X$, where $J$ and $K$ are the corresponding index sets. We say $\xi$ is a refinement of $\eta$ if for each $A_j \in \xi$, there exists $B_k \in \eta$ such that $A_j \subseteq B_k$. The common refinement $\xi \vee \eta$ of $\xi$ and $\eta$ defined as
$$\xi \vee \eta=\left{A_j \cap B_k \mid j \in J, k \in K\right}$$
is also a cover. Note that if $\xi$ and $\eta$ are both open covers (resp., measurable partitions), then $\xi \vee \eta$ is also an open cover (resp., a measurable partition). Define $g^{-1}(\xi)=$ $\left{g^{-1}\left(A_j\right) \mid j \in J\right}$, and denote for $n \in \mathbb{N}$,
$$\xi_g^n=\bigvee_{j=0}^{n-1} g^{-j}(\xi)=\xi \vee g^{-1}(\xi) \vee \cdots \vee g^{-(n-1)}(\xi),$$
and let $\xi_g^{\infty}$ be the smallest $\sigma$-algebra containing $\bigcup^{+\infty} \xi_g^n$.

## 数学代写|遍历理论代考Ergodic theory代考|Entropy and Pressure

Let $(X, d)$ be a compact metric space and $g: X \rightarrow X$ a continuous map. For $n \in \mathbb{N}$ and $x, y \in X$,
$$d_g^n(x, y)=\max \left{d\left(g^k(x), g^k(y)\right) \mid k \in{0,1, \ldots, n-1}\right}$$
defines a new metric on $X$. A set $F \subseteq X$ is $(n, \varepsilon)$-separated, for some $n \in \mathbb{N}$ and $\varepsilon>0$, if for each pair of distinct points $x, y \in F$, we have $d_g^n(x, y) \geq \varepsilon$. For $\varepsilon>0$ and $n \in \mathbb{N}$, let $F_n(\varepsilon)$ be a maximal (in the sense of inclusion) $(n, \varepsilon)$-separated set in $X$.

For each $\psi \in C(X)$, the following limits exist and are equal, and we denote the limits by $P(g, \psi)$ (see for example, [PU10, Theorem 3.3.2]):

\begin{aligned} P(g, \psi) &=\lim {\varepsilon \rightarrow 0} \limsup {n \rightarrow+\infty} \frac{1}{n} \log \sum_{x \in F_n(\varepsilon)} \exp \left(S_n \psi(x)\right) \ &=\lim {\varepsilon \rightarrow 0} \liminf {n \rightarrow+\infty} \frac{1}{n} \log \sum_{x \in F_n(\varepsilon)} \exp \left(S_n \psi(x)\right), \end{aligned}
where $S_n \psi(x)=\sum_{j=0}^{n-1} \psi\left(g^j(x)\right)$ is defined in $(0.3)$. We call $P(g, \psi)$ the topological pressure of $g$ with respect to the potential $\psi$. The quantity $h_{\text {top }}(g)=P(g, 0)$ is called the topological entropy of $g$. Note that $P(g, \psi)$ is independent of $d$ as long as the topology on $X$ defined by $d$ remains the same (see [PU10, Sect. 3.2]).
We now review measure-theoretic counterparts of the concepts above.
The information function I maps a measurable partition $\xi$ of $X$ to a $\mu$-a.e. defined real-valued function on $X$ in the following way:
$$I(\xi)(x)=-\log \mu(\xi(x)), \quad \text { for } x \in X$$

## 数学代写|遍历理论代考Ergodic theory代考|Covers and Partitions

〈left 缺少或无法识别的分隔符

$$\xi_g^n=\bigvee_{j=0}^{n-1} g^{-j}(\xi)=\xi \vee g^{-1}(\xi) \vee \cdots \vee g^{-(n-1)}(\xi),$$

## 数学代写|遍历理论代考Ergodic theory代考|Entropy and Pressure

〈left 缺少或无法识别的分隔符

$$P(g, \psi)=\lim \varepsilon \rightarrow 0 \lim \sup n \rightarrow+\infty \frac{1}{n} \log \sum_{x \in F_n(\varepsilon)} \exp \left(S_n \psi(x)\right) \quad=\lim \varepsilon \rightarrow 0 \liminf n \rightarrow+\infty \frac{1}{n} \log \sum_{x \in F_n(\varepsilon)} \exp \left(S_n \psi(x)\right),$$

$$I(\xi)(x)=-\log \mu(\xi(x)), \quad \text { for } x \in X$$

## 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。