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# 物理代写|统计力学代写Statistical Mechanics代考|PHYS602 Statistical Theory of Dynamical Systems

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## 物理代写|统计力学代写Statistical Mechanics代考|Statistical Theory of Dynamical Systems

Given a system with sensitive dependence on initial conditions, what can one do? We have seen that one cannot predict trajectories beyond a certain “temporal horizon”. The next best thing one can try to do is to predict statistical properties of the trajectories of that system, which is similar to what one does in statistical mechanics. For example, one can try to compute the average time $\tau_{A}$ spent in a region $A \in \Sigma$, given by (4.3.8) with $F=\mathbb{1}_{A}$. But we already know how to do that: use the ergodic theorem, at least for ergodic transformations.

For example, for the maps $T$ defined by (4.1.3), (4.1.7), or (4.1.11), that are sensitive with respect to initial conditions, but also ergodic with respect to the Lebesgue measure, the average time $\tau_{A}$ spent in a region $A \in \Sigma$ is given, see (4.3.9), by $\tau_{A}=\mu_{\mathrm{Leb}}(A)$

Or consider the Bernoulli shift on $k$ symbols defined by (4.1.12), and ask with which frequency does a given finite string of symbols $\left(\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n-1}\right)$, with $\alpha_{i} \in$ ${0,1, \ldots, k-1}, \forall i=0,1, \ldots, n-1$ occur in an element $\mathbf{x} \in \boldsymbol{\Omega}{k}$ ? Obviously any frequency will occur for some element $\mathbf{x} \in \boldsymbol{\Omega}{k}$, because we can simply construct such an element by inserting the finite sequence $\left(\alpha_{0}, \alpha_{1} \ldots, \alpha_{n-1}\right)$ in the infinite sequence $\mathbf{x}$ with the desired frequency.

But if one asks the same question for almost all $\mathbf{x}$ with respect to to the product measure $\boldsymbol{\mu}$ on $\boldsymbol{\Omega}{k}$, then there is a unique answer. Let $A{\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n-1}}={\mathbf{x}=$ $\left.\left(x_{n}\right){n \in \mathbb{Z}}, x{0}=\alpha_{0}, x_{1}=\alpha_{1}, \ldots, x_{n-1}=\alpha_{n}\right}$. Then the frequency of appearance of the sequence $\left(\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n-1}\right)$ in $\mathbf{x} \in \boldsymbol{\Omega}{k}$ is given by: $$\lim {N \rightarrow \infty} \frac{1}{N} \sum_{m=0}^{N-1} \mathbb{1}{A{a_{0}, a_{1}, \ldots, a_{n-1}}}\left(T_{\text {shift }}^{m} \mathbf{x}\right)$$
with $T_{\text {shift }}$ defined in (4.1.12); by the ergodic theorem the limit in (4.6.1) exists for almost all $\mathbf{x}$ with respect to to the product measure $\boldsymbol{\mu}$ on $\boldsymbol{\Omega}{k}$, and, by ergodicity of $T{\text {shift }}$, is equal to $\boldsymbol{\mu}\left(A_{\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n-1}}\right)=\prod_{i=0}^{n-1} \mu\left(\alpha_{i}\right)$.

## 物理代写|统计力学代写Statistical Mechanics代考|Itineraries and Coding

An important tool in the study of dynamical systems is the notion of coding, which means a correspondance between a trajectory of a dynamical system and a sequence of symbols. Let $T$ be a $\mu$ invariant map on $\Omega$ and let $\left(\Omega_{0}, \Omega_{1}, \ldots, \Omega_{k-1}\right)$ be a partition of $\Omega: \Omega=\cup_{i=0}^{k-1} \Omega_{i}, \Omega_{i} \cap \Omega_{j}=\emptyset, i \neq j$. Let us set $k=2$ for simplicity.

Given $x \in \Omega$, one defines the following map from $\sigma: \Omega \rightarrow{0,1}^{\mathbb{N}}=\boldsymbol{\Omega}{2}^{+}$: $$\begin{gathered} \sigma(x){n}=0 \text { if } \quad T^{n}(x) \in \Omega_{0} \ \sigma(x){n}=1 \text { if } \quad T^{n}(x) \in \Omega{1} . \end{gathered}$$

## 物理代写|统计力学代写Statistical Mechanics代考|Statistical Theory of Dynamical Systems

$$\lim N \rightarrow \infty \frac{1}{N} \sum_{m=0}^{N-1} 1 A a_{0}, a_{1}, \ldots, a_{n-1}\left(T_{\text {shift }}^{m} \mathbf{x}\right)$$ $T$ shift ,等于 $\boldsymbol{\mu}\left(A_{\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n-1}}\right)=\prod_{i=0}^{n-1} \mu\left(\alpha_{i}\right)$.

## 物理代写|统计力学代写Statistical Mechanics代考|ltineraries and Coding

$$\sigma(x) n=0 \text { if } \quad T^{n}(x) \in \Omega_{0} \sigma(x) n=1 \text { if } \quad T^{n}(x) \in \Omega 1 .$$

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