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

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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

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## 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。