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# 数学代写|遍历理论代考Ergodic theory代考|MATH6720 Equilibrium States

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## 数学代写|遍历理论代考Ergodic theory代考|Equilibrium States

In this chapter, we investigate the existence, uniqueness, and other properties of equilibrium states for an expanding Thurston map. The main tool for this chapter is the thermodynamical formalism. We record the main theorem of this chapter below.
Theorem 5.1 Let $f: S^2 \rightarrow S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$. Let $\phi$ be a real-valued Hölder continuous function on $S^2$ with respect to the metric $d$.

Then there exists a unique equilibrium state $\mu_\phi$ for the map $f$ and the potential $\phi$. If $\psi$ is another real-valued Hölder continuous function on $S^2$ with respect to the metric $d$, then $\mu_\phi=\mu_\psi$ if and only if there exists a constant $K \in \mathbb{R}$ such that $\phi-\psi$ and $K \mathbb{1}{S^2}$ are co-homologous in the space of real-valued continuous functions on $S^2$, i.e., $\phi-\psi-K \mathbb{1}{S^2}=u \circ f-u$ for some real-valued continuous function $u$ on $S^2$.
Moreover, $\mu_\phi$ is a non-atomic $f$-invariant Borel probability measure on $S^2$ and the measure-preserving transformation $f$ of the probability space $\left(S^2, \mu_\phi\right)$ is forward quasi-invariant, nonsingular, exact, and in particular, mixing and ergodic.

In addition, the preimages points of $f$ are equidistributed with respect to $\mu_\phi$, i.e., for each sequence $\left{x_n\right}_{n \in \mathbb{N}}$ of points in $S^2$, as $n \longrightarrow+\infty$,
$$\begin{gathered} \frac{1}{Z_n(\phi)} \sum_{y \in f^{-n}\left(x_n\right)} \operatorname{deg}{f^n}(y) \exp \left(S_n \phi(y)\right) \frac{1}{n} \sum{i=0}^{n-1} \delta_{f^i(y)} \stackrel{w^}{\longrightarrow} \mu_\phi, \ \frac{1}{Z_n(\widetilde{\phi})} \sum_{y \in f^{-n}\left(x_n\right)} \operatorname{deg}{f^n}(y) \exp \left(S_n \widetilde{\phi}(y)\right) \delta_y \stackrel{w^}{\longrightarrow} \mu\phi, \end{gathered}$$
where $Z_n(\psi)=\sum_{y \in f^{-n}\left(x_n\right)} \operatorname{deg}_{f^n}(y) \exp \left(S_n \psi(y)\right)$, for each $n \in \mathbb{N}$ andeach $\psi \in C\left(S^2\right)$.

## 数学代写|遍历理论代考Ergodic theory代考|The Assumptions

We state below the hypothesis under which we will develop our theory in most parts of this chapter and Chaps. 6 and 7. We will repeatedly refer to such assumptions in these chapters. We emphasize again that not all assumptions are assumed in all the statements in the subsequent chapters, and that in fact we have to gradually remove the dependence on some of the assumptions before establishing our main results.
The Assumptions

1. $f: S^2 \rightarrow S^2$ is an expanding Thurston map.
2. $\mathscr{C} \subseteq S^2$ is a Jordan curve containing post $f$ with the property that there exists $n_{\mathscr{C}} \in \mathbb{N}$ such that $f_{\mathscr{C}}(\mathscr{C}) \subseteq \mathscr{C}$ and $f^m(\mathscr{C}) \nsubseteq \mathscr{C}$ for each $m \in\left{1,2, \ldots, n_{\mathscr{C}}-\right.$ $1}$.
3. $d$ is a visual metric on $S^2$ for $f$ with expansion factor $\Lambda>1$ and a linear local connectivity constant $L \geq 1$.
4. $\phi \in C^{0, \alpha}\left(S^2, d\right)$ is a real-valued Hölder continuous function with an exponent $\alpha \in(0,1]$

Observe that by Theorem 2.16, for each $f$ in (1), there exists at least one Jordan curve $\mathscr{C}$ that satisfies (2). Since for a fixed $f$, the number $n \mathscr{C}$ is uniquely determined by $\mathscr{C}$ in (2), in the remaining part of the paper we will say that a quantity depends on $\mathscr{C}$ even if it also depends on $n_{\mathscr{C}}$.

Recall that the expansion factor $\Lambda$ of a visual metric $d$ on $S^2$ for $f$ is uniquely determined by $d$ and $f$. We will say that a quantity depends on $f$ and $d$ if it depends on $\Lambda$.

Note that even though the value of $L$ is not uniquely determined by the metric $d$, in the remainder of this paper, for each visual metric $d$ on $S^2$ for $f$, we will fix a choice of linear local connectivity constant $L$. We will say that a quantity depends on the visual metric $d$ without mentioning the dependence on $L$, even though if we had not fixed a choice of $L$, it would have depended on $L$ as well.

In the discussion below, depending on the conditions we will need, we will sometimes say “Let $f, \mathscr{C}, d, \phi, \alpha$ satisfy the Assumptions.”, and sometimes say “Let $f$ and $d$ satisfy the Assumptions.”, etc.

## 数学代写|遍历理论代考Ergodic theory代考|The Assumptions

1. $f: S^2 \rightarrow S^2$ 是一个扩展的瑟斯顿地图。
2. $\mathscr{C} \subseteq S^2$ 是一条包含柱子的乔丹曲线 $f$ 具有存在的属性 $n \mathscr{C} \in \mathbb{N}$ 这样 $f_{\mathscr{C}}(\mathscr{C}) \subseteq \mathscr{C}$ 和 $f^m(\mathscr{C}) \nsubseteq \mathscr{C}$ 每个
〈left 缺少或无法识别的分隔符
3. $\phi \in C^{0, \alpha}\left(S^2, d\right)$ 是具有指数的实值 Hölder 连紏函数 $\alpha \in(0,1]$
观察定理 2.16，对于每个 $f(1)$ 中，至少存在一条若尔当曲线 $\mathscr{C}$ 满足 (2) 。因为对于一个固定的 $f$ ，号码 $n \mathscr{C}$ 唯一地由 $\mathscr{C}$ 在 (2) 中，在本文的其余部分，我们将涚数量取决于 $\mathscr{C}$ 即使它也取决于 $n \mathscr{E}$.
回想一下扩展因子 $\Lambda$ 视觉指标 $d$ 上 $S^2$ 为了 $f$ 唯一地由 $d$ 和 $f$. 我们会说数量取决于 $f$ 和 $d$ 如果这取决于 $\Lambda$.
请注意，即使值 $L$ 不是由度量唯一确定的 $d$ ，在本文的其余部分，对于每个视觉指标 $d$ 上 $S^2$ 为了 $f$ ，我们将修复线性局部连接営数 的选择 $L$. 我们会说数量取决于视觉指标 $d$ 更不用说依赖 $L$ ，即使我们没有固定选择 $L$ ，这将取决于 $L$ 以及。
在下面的讨论中，根据我们需要的条件，我们有时会说”让 $f, \mathscr{C}, d, \phi, \alpha$ 满足假设。”，有时会说”让 $f$ 和 $d$ 满足假设。”等。

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