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## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|A little bit of measure theory

a. A little bit of measure theory. As we have seen, computing the Hausdorff dimension of a set can be very difficult, even if the set is geometrically quite regular, as was the case for the Cantor sets we have considered so far. For sets whose self-similarity is not quite so regular, the situation becomes even worse; for example, consider the non-linear map shown in Figure 1.23. This map generates a repelling Cantor set $C$ through the same construction that we carried out for the linear map in Figure 1.12, and we may ask what the Hausdorff dimension of $C$ is. Thus we must study a Cantor construction in which the ratio coefficients are no longer constant but may change at each step of the iteration. This occurs because the map is no longer linear, and so how much contraction (in the construction) or expansion (in the map) occurs at each step now depends on which point we consider, not just on which interval it is in.

How does the Hausdorff dimension respond to this change in the construction? The problems which arise at this stage are much more difficult than those we encountered in proving Theorem 2.21, and we will need new tools to deal with them.

A key idea will be to somehow sidestep the fact that ratio coefficients may vary by studying “asymptotic ratio coefficients” which give the average rate of contraction over a large number of steps of the construction. If this average rate converges, we can still hope to say something; however, at some point it will become necessary to distinguish between “bad” points of $C$, which we want to ignore because the average rate does not converge, and “good” points, which we can deal with, and we will need to show that there are in some sense “more” of the latter.

In order to make all this precise, we need to expand our toolkit to include the idea of a measure. Without further ado, then, we have the following definition.

## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Lebesgue measure and outer measures

b. Lebesgue measure and outer measures. We now introduce a more complicated example of a measure space-Lebesgue measure on $\mathbb{R}^d$, which we first mentioned in the proof of Theorem $2.20$. In the case $d=1$, which we will consider first, this generalises the idea of “length” to apply to a broader class of sets than merely intervals. For $d=2$, it generalises area; for $d=3$, it generalises volume; and for $d \geq 4$, it generalises $d$-dimensional volume.

The full construction of Lebesgue measure is one of the primary parts of measure theory, and a complete treatment of all the details requires most of a graduate-level course, so our discussion here will necessarily be somewhat abbreviated, and we will omit proofs. ${ }^1$

In order to construct a measure space $(X, \mathcal{A}, m)$, we must do two things. First, we must produce a $\sigma$-algebra $\mathcal{A}$, in which we would like to include as many sets as possible, to make our measure as useful as possible. Second, we must figure out how to define a set function $m$ which satisfies the three properties of a measure. In particular, it is far from clear how to guarantee that whatever function $m$ we construct is $\sigma$-additive, especially if the collection $\mathcal{A}$ of sets for which this must be checked is very large.

# 分形几何和混沌系统代考

## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Lebesgue measure and outer measures

Lebesgue测度的完整构造是测度论的主要部分之一，所有细节的完整处理需要大部分研究生水平的课程，因此我们这里的讨论必然会有所简化，并且我们将省略证明。1

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

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## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Comparison of Hausdorﬀﬀ and topological dimension

a. Comparison of Hausdorff and topological dimension. One difference in the two dimensions we have defined is immediately apparent; the topological dimension is always an integer, while the Hausdorff dimension has no a priori reason to take integer values. Indeed, it can take any non-negative real value (see Exercise 2.20).

Another difference becomes apparent if we look at what notions are used in the definitions; the topological dimension can be defined for any topological space, whether or not it has a metric, while the Hausdorff dimension requires a metric for its definition. If we need to explicitly indicate the metric being used, we will write the Hausdorff dimension of $Z$ with respect to the metric $d$ as $\operatorname{dim}_H^d Z$.

This distinction becomes important when we observe that a single topological space can be equipped with multiple metrics. For example, the usual metric on $\mathbb{R}^d$ is given by Pythagoras’ formula
$$d(x, y)=\sqrt{\sum_i\left(x_i-y_i\right)^2},$$
but other metrics may be introduced by the formulae
\begin{aligned} \rho(x, y) &=\sum_i\left|x_i-y_i\right|, \ \sigma(x, y) &=\max _i\left|x_i-y_i\right|, \end{aligned}
and it is not hard to check that these metrics all induce the same topology on $\mathbb{R}^d$ (see Exercise 2.5). In particular, they all lead to the same topological dimension; do they all lead to the same Hausdorff dimension? To answer this question, we need some new definitions, giving three senses in which two metrics $d_1$ and $d_2$ on $\mathbb{R}^d$ (or more generally, on any metric space $X$ ) may be said to be “the same”.

## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Metrics and topologies

b. Metrics and topologies. So far, all our examples of topological spaces have been metric spaces as well. One may rightly ask, then, if every example arises this way; given a topological space $(X, \mathcal{T})$, can we always find a metric $d$ on $X$ such that the sets in $\mathcal{T}$ are precisely those sets which are unions of $d$-balls? Such a space is called metrisable, and so we may ask, are all topological spaces metrisable?
It turns out that the answer is “no”: Some topologies do not come from metrics. But which ones? Given a particular topology, how can we tell whether or not it comes from a metric? To answer this question, we examine properties of metric spaces which do not follow from the axioms of a topological space.

Exercise 2.10. Let $(X, d)$ be a metric space, and fix $x \in X$. Show that the set ${x}$ is closed.

Exercise 2.11. Let $(X, d)$ be a metric space, and fix $x, y \in X$. Show that there exist disjoint open sets $U, V \subset X$ such that $x \in U$ and $y \in V$; that is, metric spaces are Hausdorff.

Exercise 2.12. Let $(X, d)$ be a metric space, and let $A, B \subset X$ be disjoint closed sets. Show that there exist disjoint open sets $U, V \subset X$ such that $A \subset U$ and $B \subset V$; that is, metric spaces are normal.

# 分形几何和混沌系统代考

## 数学代写|分形几何和混沌系统代考Fractal Geometry \& Chaotic Dynamics代 写|Comparison of Hausdorffff and topological dimension

$$d(x, y)=\sqrt{\sum_i\left(x_i-y_i\right)^2},$$

$$\rho(x, y)=\sum_i\left|x_i-y_i\right|, \sigma(x, y) \quad=\max _i\left|x_i-y_i\right|$$

## 数学代写|分形几何和混沌系统代考Fractal Geometry \& Chaotic Dynamics代 与|Metrics and topologies

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Fractal Geometry & Chaotic Dynamics, 分形几何和混沌系统, 数学代写

## avatest™帮您通过考试

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## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|The topological structure of symbolic space and the Cantor set

a. The topological structure of symbolic space and the Cantor set. Having seen that $h$ respects the dynamics of $f$ and $\sigma$, it is natural to ask what other aspects of the sets $C$ and $\Sigma_2^{+}$are preserved by the conjugacy. We saw in the previous lecture that the Cantor set $C$ inherits a metric, and hence a topology, from the real line. If we define a metric on $\Sigma_2^{+}$, then we will have a topology there too, and we may ask whether $h$ takes convergent sequences in $\Sigma_2^{+}$to convergent sequences in $C$, and vice versa. This will expand the range of questions about the dynamics of $f$ which can be answered by looking at the symbolic case to include questions of a topological nature – that is, questions involving convergence.

To this end, fix a real number $a>1$, and given two sequences $v, w \in \Sigma_2^{+}$, define the distance between them by
$$d_a(v, w)=\sum_{k \geq 1} \frac{\left|v_k-w_k\right|}{a^j} .$$
Note that since each numerator $\left|v_k-w_k\right|$ is either 0 or 1 , this series converges absolutely. We may easily verify that $d=d_a$ satisfies the axioms of a metric from the previous lecture, each of which follows immediately from its counterpart for the usual distance on $\mathbb{R}$.

## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|What the coding map doesn’t do

b. What the coding map doesn’t do. Despite the fact that the map $f$ and the shift $\sigma$ are topologically conjugate, the two systems are not equivalent in every aspect; $\Sigma_2^{+}$does not capture quite everything there is to know about the Cantor set $C$. To convince ourselves of this, let us consider a more general class of dynamical systems defined in the interval. Fix two disjoint closed intervals $I_1, I_2 \subset[0,1]$, and define a piecewise linear map $f: I_1 \cup I_2 \rightarrow[0,1]$ as shown in Figure 1.17, so that $f\left(I_1\right)=f\left(I_2\right)=[0,1]$ (note that for our purposes, each branch of $f$ may be either increasing or decreasing).

If we try to iterate $f$ more than once, we run into the same problem as before; some points in $I_1$ or $I_2$ have images which do not lie in either interval, and so cannot be iterated again. This leads us down exactly the same path as in Lecture 3; the domain of definition of $f^2$ is a union of four intervals, as shown in Figure 1.18, and so on for $f^3, f^4, \ldots$. The only difference in this case is that the intervals may be of varying lengths, but the combinatorial and topological structure is identical to that in the previous analysis, and we again get $a$ Cantor set (rather than the middle-third Cantor set), for which we have a coding map and symbolic dynamics just as before.

# 分形几何和混沌系统代考

## 数学代写|分形几何和混沌系统代考Fractal Geometry \& Chaotic Dynamics代 写|The topological structure of symbolic space and the Cantor set

$$d_a(v, w)=\sum_{k \geq 1} \frac{\left|v_k-w_k\right|}{a^j} .$$

## 数学代写|分形几何和混沌系统代考Fractal Geometry \& Chaotic Dynamics代 写|What the coding map doesn’t do

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。