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数学代写|遍历理论代考Ergodic theory代考|MA795 Thurston Maps

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数学代写|遍历理论代考Ergodic theory代考|Thurston Maps

In this chapter, we introduce the dynamical systems that we are going to study, namely, expanding Thurston maps. We first recall briefly some history in Sect. 2.1, where we by no means intend to give a complete account of the development of the subject. We then introduce Thurston maps in Sect. $2.2$ and certain cell decompositions of the 2-sphere $S^2$ induced by Thurston maps in Sect. 2.3. Next, we discuss various notions of expansion in our context and define expanding Thurston maps in Sect. 2.4. Two most important tools in the study of expanding Thurston maps are explored in the last two sections. The first tool is a natural class of metrics on the $S^2$, called visual metrics, discussed in Sect. 2.5. The second tool, discussed in Sect. 2.6, is the existence and properties of certain forward invariant Jordan curves on $S^2$, which induce nice partitions of the sphere. It is the geometric and combinatorial information we get from these tools that enables us to investigate the dynamical properties of expanding Thurston maps.

We prove in Lemma $2.12$ that the union of all iterated preimages of an arbitrary point $p \in S^2$ of an expanding Thurston map is dense in $S^2$. We also summarize properties of visual metrics from [BM17], especially the relation between visual metrics and the cell decompositions, in Lemma $2.13$ and the discussion that follows it. The fact that an expanding Thurston map is Lipschitz with respect to a visual metric is established in Lemma 2.15. M. Bonk and D. Meyer proved that for each expanding Thurston map $f$, there exists an $f^n$-invariant Jordan curve containing post $f$ for each sufficiently large $n \in \mathbb{N}$ depending on $f$ (see Theorem 2.16). We prove in Lemma $2.17$ a slightly stronger version of this result, which carries additional combinatorial information of the Jordan curve. This lemma will be used in Chaps. 4 and 6. Finally, in Lemma 2.19, we prove that an expanding Thurston map locally expands the distance, with respect to a visual metric, between two points exponentially as long as they belong to one set in some particular partition of $S^2$ induced by a backward iteration of some Jordan curve on $S^2$. This observation, generalizing a result of M. Bonk and D. Meyer [BM17, Lemma 15.25], enables us to establish the distortion lemmas (Lemmas $5.3$ and 5.4) in Sect. 5.2, which serve as the cornerstones for the mechanism of thermodynamical formalism that is essential in Chap. 5 .

数学代写|遍历理论代考Ergodic theory代考|Historical Background

The study of Thurston maps dates back to W.P. Thurston’s celebrated combinatorial characterization theorem of postcritically-finite rational maps on the Riemann sphere among a class of more general continuous maps [DH93]. We call this class of continuous maps Thurston maps nowadays. Thurston’s theorem asserts that a Thurston map is essentially a rational map if and only if there does not exist a so-called Thurston obstruction, i.e., a collection of simple closed curves on $S^2$ subject to certain conditions [DH93]. Due to the important and fruitful applications of Thurston’s theorem, many authors have worked on extending it beyond postcritically-finite rational maps using similar combinatorial obstructions. See for example, J.H. Hubbard, D. Schleicher, M. Shishikura’s work on some postcritically-finite exponential maps [HSS09]; G. Cui and L. Tan’s and G. Zhang and Y. Jiang’s works on hyperbolic rational maps [CT11, ZJ09]; G. Zhang’s work on certain rational maps with Siegel disks [Zh08]; X. Wang’s work on certain rational maps with Herman rings [Wan 14]; and G. Cui and L. Tan’s work on some geometrically finite rational maps [CT15].

It has since been a central theme in the study of conformal dynamical systems to search for Thurston-type theorems, i.e., characerizations of conformal dynamical systems in a wider class of dynamical systems satisfying suitable metric-topological conditions. See also [Th16, KPT15] for some remarkable recent works in this direction.

It is natural to ask for Thurston-type theorems from different points of view. One promising approach is from a point of view of metric space properties. In order to gain more precise metric estimates, groups of authors, notably M. Bonk and D. Meyer [BM17], P. Haïssinsky and K. Pilgrim [HP09], and J.W. Cannon, W.J. Floyd, and R. Parry [CFP07] started to impose natural notions of expansion in their respective contexts. These notions turned out to coincide in the context of expanding Thurston maps (see Sect. $2.4$ for more details).

The existence of certain invariant Jordan curves as stated in Theorem $2.16$ serves as foundation and starting point of the investigation of expanding Thurston maps. The special case of Theorem $2.16$ for rational expanding Thurston maps was announced by M. Bonk during an Invited Address at the AMS Meeting at Athens, Ohio, in March 2004, where W.J. Floyd also mentioned a related result independently obtained by J.W. Cannon, W.J. Floyd, and R. Parry [CFP07]. Finally a Thurston-type theorem from a metric space point of view was obtained independently by M. Bonk and D. Meyer [BM17], and P. Haïssinsky and K. Pilgrim [HP09] in their respective contexts. See Theorem $1.1$ in the case of expanding Thurston maps. Special cases of Theorem $1.1$ go back to [Me02] and unpublished joint work of M. Bonk and B. Kleiner (see [BM17, Preface]).

数学代写|遍历理论代考Ergodic theory代考|Historical Background

Thurston 映射的研究可以追溯到 WP Thurston 在一类更一般的连续映射 [DH93] 中黎曼球面上的后临界有限有理映射的著名组合表征定理。现在我们称这类连续映射为 Thurston 映射。瑟斯顿定理断言瑟斯顿映射本质上是有理映射当且仅当不存在所谓的瑟斯顿障碍，即简单闭合曲线的集合小号2受某些条件限制 [DH93]。由于瑟斯顿定理的重要和富有成效的应用，许多作者致力于使用类似的组合障碍将其扩展到后临界有限有理图之外。例如，参见 JH Hubbard、D. Schleicher、M. Shishikura 关于一些后临界有限指数映射的工作 [HSS09]；G. Cui 和 L. Tan 以及 G. Zhang 和 Y. Jiang 关于双曲有理映射的著作 [CT11, ZJ09]；G. Zhang 在某些带有 Siegel 圆盘的有理映射上的工作 [Zh08]；X. 王在某些带赫尔曼环的有理图上的工作[Wan 14]；G. Cui 和 L. Tan 关于一些几何有限有理图的工作 [CT15]。

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