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## 物理代写|统计力学代写Statistical Mechanics代考|The Solenoid

The reader might worry that, in the previous example, the map $T$ is not continuous. That is why we will also consider the example of the solenoid, which is closely related to the modified baker’s map, but where this problem is avoided. We will follow Lanford [207] in the presentation of this example.

Let $\mathbf{T}$ be a solid torus in three dimensions: $\mathbf{T}=S^{1} \times D^{2}$, with $S^{1}$ the unit circle, $S^{1}={z \in \mathbb{C},|z|=1}$ and $D^{2}$ the unit disk $D^{2}={w \in \mathbb{C},|w| \leq 1}$. We define the solenoid map:
$$T(z, w)=\left(z^{2}, \frac{1}{2} z+\frac{1}{4} w\right)$$
where, if we write $z=\exp (2 \pi i x)$, we see that $z \rightarrow z^{2}$ is just the map $x \rightarrow 2 x$ mod 1 defined in (4.1.3). The image of the torus under $T$ is a tube inside the torus of transverse radius $\frac{1}{4}$ winding twice around the torus, see Fig. 4.5. The intersection of this tube with the transverse plane ${z=1}$ is made of two disjoint disk, each of radius $\frac{1}{4}$. Iterating, we see that the image of the torus under $T^{n}$ is a tube inside the torus of transverse radius $\frac{1}{4^{n}}$ winding $2^{n}$ times around the torus. The intersection of this tube with the transverse plane $z=1$ is made of $2^{n}$ disjoint disks, each of radius $\frac{1}{4^{n}}$.

One defines the solenoid as $\Lambda=\bigcap_{n \in \mathbb{N}} T^{n}(\mathbf{T})$. It is composed of a set of lines (with no thickness) wrapping around the torus. Its intersection with the transverse plane ${z=1}$ is a Cantor set. Since that set is uncountable and each line in the solenoid has only countably many intersections with ${z=1}$, the solenoid is not a single line but an uncountable union of lines. Cutting the solenoid along the ${z=1}$ plane splits $\Lambda$ into uncountably many loops going once around the torus, that can be labelled by points in a Cantor set.

## 物理代写|统计力学代写Statistical Mechanics代考|The Logistic Map

Let us admit that the tent map (4.1.8) is ergodic with respect to the Lebesgue measure (this is true but not very easy to prove). One can check, by explicit computation that, if we denote the logistic map $4 x(1-x)=g(x)$, and write $x=C(y)$, with:
$$C(y)=\frac{1-\cos \pi y}{2}$$
Then, $g(x)=C \circ f \circ C^{-1}(x)$ with $f$ the tent map: indeed, $g(C(y))=4\left(\frac{1-\cos \pi y}{2}\right)$ $\left(\frac{1+\cos \pi y}{2}\right)=\sin ^{2} \pi y$, and, for $0 \leq y \leq \frac{1}{2}, C(f(y))=\frac{1-\cos 2 \pi y}{2}=\sin ^{2} \pi y$, while for $\frac{1}{2} \leq y \leq 1, C(f(y))=\frac{1-\cos 2 \pi(1-y)}{2}=\sin ^{2} \pi y$.

So, $g^{n}(x)=C \circ f^{n} \circ C^{-1}(x)$ and, if one wants to compute the average time spent by the orbit of $x$ in an interval $J$ for the map $g$ :
$$\lim {N \rightarrow \infty} \frac{1}{N} \sum{n=0}^{N-1} I\left(g^{n}(x) \in J\right)$$
it is enough to compute
$$\lim {N \rightarrow \infty} \frac{1}{N} \sum{n=0}^{N-1} I\left(f^{n}(y) \in C^{-1}(J)\right)$$

## 物理代写|统计力学代写Statistical Mechanics代考|The Solenoid

$$T(z, w)=\left(z^{2}, \frac{1}{2} z+\frac{1}{4} w\right)$$

## 物理代写|统计力学代写Statistical Mechanics代考|The Logistic Map

$$C(y)=\frac{1-\cos \pi y}{2}$$

$$\lim N \rightarrow \infty \frac{1}{N} \sum n=0^{N-1} I\left(g^{n}(x) \in J\right)$$

$$\lim N \rightarrow \infty \frac{1}{N} \sum n=0^{N-1} I\left(f^{n}(y) \in C^{-1}(J)\right)$$

## 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。