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One Dimensional Dynamical System, Singapore

在新加坡的这五年—学术篇

April 5, 2020 zr9558 Leave a comment

每次提到数学这个词，大家能够想到的就是初等代数，平面几何，组合运算，微积分，线性代数，概率论等方向。但在整个数学领域（Earth of Math）上，还有很多更有意思的领域和研究方向，包括数论，几何，拓扑，分形几何，分析，概率统计，博弈论，代数等诸多方向，每一个方向都有很多优秀的数学家在从事相关研究。

当年在数学系的时候，所研究的方向是分形几何（Fractal Geometry）和复动力系统（Complex Dynamics），位于 Earth of Math 的左侧，称之为分形湖泊（Fractal Lakes）。所谓分形，其实是一个粗糙或者零碎的集合形状，可以分成多个部分，且每个部分放大之后与整体有某种相似性，即具有自相似性的性质。而动力系统则是基于某种固定的规则，描述一个空间内的所有点随时间的变化情况，例如钟摆的晃动，水的流动，湖泊里面鱼类的数量。备注：动力系统并不是指汽车的动力系统和发动机引擎，这两者毫无关系。

而复动力系统则是动力系统中的一个分支，研究的是有理函数的迭代性质。所谓函数的迭代，指的是针对有理函数 $f(z)$ ，考察其定义域的点 $z\in \hat{\mathbb{C}}$ 的 $n$ 次复合，得到 $f^{(n)}(z)=f\circ\cdots\circ f(z)$ ，进一步可以研究 $\lim_{n\rightarrow +\infty}f^{(n)}(z)$ 的极限。

针对不同的定义域，函数的迭代有着完全不同的研究方法。当时的研究方向是复动力系统（Complex Dynamical Systems）。复动力系统理论的研究始于 1920 年，当时是由数学家 Fatou 和 Julia 研究的，因此复动力系统中的两个重要的集合就是以 Fatou 和 Julia 来命名的，分别称之为 Fatou set 和 Julia Set。随着计算机技术的演进，在上世纪八十年代这些集合可以通过计算机进行可视化，分形几何和复动力系统理论开始蓬勃发展起来。在与双曲几何、分形几何、现代分析学和混沌学等学科发展相互促进的同时，围绕双曲猜想以及 Mandelbrot 集合的研究工作，成为当今复动力系统的研究热点。

举个例子，函数 $f(z) = z^{2}+0.7885 e^{ia}$ （ $a\in[0,2\pi]$ ）的 Julia 集合的动图如下：

当然，科研的时候可不是做一点可视化就算完成任务了，还是需要按部就班的学习各种数学知识和技能。

之前在学校研究动力系统的时候，收集过一些书籍，在此列举给大家，希望对初学者有一定的帮助。One Dimensional Real and Complex Dynamics（实与复动力系统）需要学习的资料如下：

基础书籍：

复分析基础：本科生课程。学习数学知识自然需要循序渐进，除了必要的数学分析，高等代数之外，分析学则是动力系统所必备的知识之一。既然是复动力系统，那肯定就要集中于研究复分析，因此本科的复分析则是复动力系统的必修课之一。

(1) Complex Analysis, 3rd Edition, Lars V. Ahlfors

(2) Complex Analysis, Elias M. Stein

进阶复分析：研究生课程

到了研究生阶段，其实也不足以直接上手搞科研，需要进一步地学习黎曼曲面，拟共形映射等专业书籍，才能够为复动力系统的学习打下基础。

(1) Lectures on Riemann Surfaces (GTM 81), Otto Forster

(2) Lectures on Quasiconformal Mappings, Lars V. Ahlfors

实分析基础：本科生课程

研究动力系统，实分析也是其基础知识之一，无论是通过学习 Stein 还是 Rudin 的教材，都是为了进一步地了解基础知识。

(1) Real Analysis and Complex Analysis, Rudin

(2) Real Analysis, Elias M. Stein

专业书籍：

实动力系统：

(1) One Dimensional Dynamics, Welington de Melo & Sebastian VanStrien

这本书难度较大，上手的时候不建议直接看这本书。

(2) Mathematical Tools for One-Dimensional Dynamics (Cambridge Studies in

Advanced Mathematics), Edson de Faria / Welington de Melo

复动力系统：

(3) Dynamics in One Complex Variable, John Milnor；Milnor 的教材总是写的清晰明确，容易上手，推荐初学者可以读这本书。

(4) Complex Dynamics, Lennart Carleson；Carleson 的教材偏向于分析学，读起来其实也有点难度，还是读 Milnor 的教材相对容易。

(5) Complex Dynamics and Renormalization, Curtis T. McMullen；McMullen 的书适合当做查阅，也不太适合从头到尾读下去。

(6) Renormalization and 3-Manifolds Which Fiber over the Circle, Curtis T. McMullen

(7) Iteration of rational functions (GTM 132), Alan F. Beardon

遍历论：

(8) An Introduction to Ergodic Theory (GTM 79), Walters Peter

学术会议

除了日常的科研之外，博士生时不时地可以去参加一下学术会议，不仅可以去参加本方向的学术会议，也可以去参加其它方向的学术会议，只要有一份邀请函即可。

如果是在 NUS 的 IMS（Institute for Mathematical Sciences）举办的学术会议，一般来说只要是在校的研究生都是可以参加的。记得当时参加的第一个学术会议是关于 PDE 的，标题叫做 Hyperbolic Conservation Laws and Kinetic Equations：Theory, Computation, and Applications（1 November – 19 December 2010）。笔者去听这个系列讲座是因为在 2010 年选择了一门 PDE 的研究生课程，而这个讲座则是作为课程的一部分。

IMSWorkshopGroupPhoto2012_6 — IMS 的偏微分方程学术会议

笔者参与的另外一个学术会议则是关于动力系统的，标题叫做 Workshop on Non-uniformly Hyperbolic and Neural One-dimensional Dynamics（23 – 27 April 2012），主要是关于非一致双曲动力系统方向的研讨会。笔者记得当时所修的课程应该只有概率论（Probability II）一门课，因此上课的任务不算很重。参会的时间恰好是学期快结束的时候，科研的任务也不算特别繁重。因此，积极参与各种学术会议也算是科研的其中一部分，一来通过参会可以了解当前的学术研究情况，二来可以认识学术界的各种人士，也算是扩大学术交流圈子的好机会。

IMSWorkshopGroupPhoto2012_2 — IMS 的动力系统学术会议

既然是学术会议，那自然就有各种各样的 Presentations，学术会议的第一天通常是需要有 IMS 的领导来致辞的，表示学术会议正式开始。每天的学术会议都需要有个 chair 来组织，一天的学术会议基本上是从早到晚，大约从早上 9:30 开始，到下午 4:40 结束。而每个学者汇报时间大约是 50 mins 左右。

这次的学术会议是关于动力系统方向的，那师兄们自然是需要上台做报告的。当时上场的师兄包括大师兄和二师兄，至于三师兄和我则暂时没有成果可以汇报。两位师兄在 IMS 的报告厅里面做了十分精彩的成果展示，会议之后也有不少同行来与师兄们讨论问题。

一般来说，每次研讨会的开始和结束都需要有一个仪式，除了 IMS 的领导致辞表示会议开始之外，在茶歇时间（Coffee Break）期间是可以四处走动的，并且在第一次茶歇的时候，全体参会人员都会在 IMS 附近拍照留念，预祝本次研讨会成功举办。

IMSWorkshopGroupPhoto2012_5 — Group Photo

论文

读到博士自然需要研究一下相应的课题，例如下面这种就是数学系博士生所研究的课题。

Question. 是否存在 $\ell\geq 4$ 的偶数和复数 $c\in\mathbb{C}$ 使得 $f(z)=z^{\ell}+c$ 的 Julia 集合 $J(f)$ 是正测度？

针对这个课题，数学系的博士生需要翻阅历史上的相关书籍和论文，阅读其相关论文才能够得到前沿技术和进展。当年花时间阅读的论文主要是几篇 Annals 上面的文章，参考资料也是这几篇文章，不过每一篇文章至少都是 40 页左右，基本上看一篇文章需要花几个月的时间。

1. Combinatorics, geometry and attractors of quasi-quadratic maps，Pages 345-404 from Volume 140 (1994), Issue 2 by Mikhail Lyubich

Paper_1

2. Wild Cantor attractors exist，Pages 97-130 from Volume 143 (1996), Issue 1 by Hendrik Bruin, Gerhard Keller, Tomasz Nowicki, Sebastian van Strien

Paper_2

3. Quadratic Julia sets with positive area，Pages 673-746 from Volume 176 (2012), Issue 2 by Xavier Buff, Arnaud Chéritat

Paper_3

4. Polynomial maps with a Julia set of positive measure，Nowicki, Tomasz, and Sebastian van Strien，arXiv preprint math/9402215(1994).

Paper_4

备注：第 4 篇文章 Polynomial maps with a Julia set of positive measure 里面有错误，通过其证明是无法得到最终结论的，因此是否存在正测度的 Julia 集合一直是未知的。直到 2012 年的第 3 篇文章出来，才算证明了二次多项式存在正测度的 Julia 集合。但是对于高次多项式，是否存在正测度的 Julia 集合则是完全未知的。

在拿到论文和课题之后，那就开始需要研究了。草稿纸也算了一张又一张，论文也打印了一份又一份，科研之路哪有一帆风顺的，基本上都是历经曲折，才能够达到毕业的彼岸。毕业的时候写了一篇文章《科研这条路》，以此来纪念读博五年的生涯。

参考资料：

科研这条路
维基百科：Julia 集合

One Dimensional Dynamical System

Hausdorff dimension of the graphs of the classical Weierstrass functions

October 10, 2016 zr9558 Leave a comment

In this paper, we obtain the explicit value of the Hausdorff dimension of the graphs of the classical Weierstrass functions, by proving absolute continuity of the SRB measures of the associated solenoidal attractors.

1. Introduction

In Real Analysis, the classical Weierstrass function is

$\displaystyle W_{\lambda,b}(x) = \sum\limits_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x)$

with ${1/b < \lambda < 1}$ .

Note that the Weierstrass functions have the form

$\displaystyle f^{\phi}_{\lambda,b}(x) = \sum\limits_{n=0}^{\infty} \lambda^n \phi(b^n x)$

where ${\phi}$ is a ${\mathbb{Z}}$ -periodic ${C^2}$ -function.

Weierstrass (1872) and Hardy (1916) were interested in ${W_{\lambda,b}}$ because they are concrete examples of continuous but nowhere differentiable functions.

Remark 1 The graph of ${f^{\phi}_{\lambda,b}}$ tends to be a “fractal object” because ${f^{\phi}_{\lambda,b}}$ is self-similar in the sense that

$\displaystyle f^{\phi}_{\lambda, b}(x) = \phi(x) + \lambda f^{\phi}_{\lambda,b}(bx)$

We will come back to this point later.

Remark 2 ${f^{\phi}_{\lambda,b}}$ is a ${C^{\alpha}}$ -function for all ${0\leq \alpha < \frac{-\log\lambda}{\log b}}$ . In fact, for all ${x,y\in[0,1]}$ , we have

$\displaystyle \frac{f^{\phi}_{\lambda, b}(x) - f^{\phi}_{\lambda,b}(y)}{|x-y|^{\alpha}} = \sum\limits_{n=0}^{\infty} \lambda^n b^{n\alpha} \left(\frac{\phi(b^n x) - \phi(b^n y)}{|b^n x - b^n y|^{\alpha}}\right),$

so that

$\displaystyle \frac{f^{\phi}_{\lambda, b}(x) - f^{\phi}_{\lambda,b}(y)}{|x-y|^{\alpha}} \leq \|\phi\|_{C^{\alpha}} \sum\limits_{n=0}^{\infty}(\lambda b^{\alpha})^n:=C(\phi,\alpha,\lambda,b) < \infty$

whenever ${\lambda b^{\alpha} < 1}$ , i.e., ${\alpha < -\log\lambda/\log b}$ .

The study of the graphs of ${W_{\lambda,b}}$ as fractal sets started with the work of Besicovitch-Ursell in 1937.

Remark 3 The Hausdorff dimension of the graph of a ${C^{\alpha}}$ -function ${f:[0,1]\rightarrow\mathbb{R}}$ is

$\displaystyle \textrm{dim}(\textrm{graph}(f))\leq 2 - \alpha$

Indeed, for each ${n\in\mathbb{N}}$ , the Hölder continuity condition

$\displaystyle |f(x)-f(y)|\leq C|x-y|^{\alpha}$

leads us to the “natural cover” of ${G=\textrm{graph}(f)}$ by the family ${(R_{j,n})_{j=1}^n}$ of rectangles given by

$\displaystyle R_{j,n}:=\left[\frac{j-1}{n}, \frac{j}{n}\right] \times \left[f(j/n)-\frac{C}{n^{\alpha}}, f(j/n)+\frac{C}{n^{\alpha}}\right]$

Nevertheless, a direct calculation with the family ${(R_{j,n})_{j=1}^n}$ does not give us an appropriate bound on ${\textrm{dim}(G)}$ . In fact, since ${\textrm{diam}(R_{j,n})\leq 4C/n^{\alpha}}$ for each ${j=1,\dots, n}$ , we have

$\displaystyle \sum\limits_{j=1}^n\textrm{diam}(R_{j,n})^d\leq n\left(\frac{4C}{n^{\alpha}}\right)^d = (4C)^{1/\alpha} < \infty$

for ${d=1/\alpha}$ . Because ${n\in\mathbb{N}}$ is arbitrary, we deduce that ${\textrm{dim}(G)\leq 1/\alpha}$ . Of course, this bound is certainly suboptimal for ${\alpha<1/2}$ (because we know that ${\textrm{dim}(G)\leq 2 < 1/\alpha}$ anyway).Fortunately, we can refine the covering ${(R_{j,n})}$ by taking into account that each rectangle ${R_{j,n}}$ tends to be more vertical than horizontal (i.e., its height ${2C/n^{\alpha}}$ is usually larger than its width ${1/n}$ ). More precisely, we can divide each rectangle ${R_{j,n}}$ into ${\lfloor n^{1-\alpha}\rfloor}$ squares, say

$\displaystyle R_{j,n} = \bigcup\limits_{k=1}^{\lfloor n^{1-\alpha}\rfloor}Q_{j,n,k},$

such that every square ${Q_{j,n,k}}$ has diameter ${\leq 2C/n}$ . In this way, we obtain a covering ${(Q_{j,n,k})}$ of ${G}$ such that

$\displaystyle \sum\limits_{j=1}^n\sum\limits_{k=1}^{\lfloor n^{1-\alpha}\rfloor} \textrm{diam}(Q_{j,n,k})^d \leq n\cdot n^{1-\alpha}\cdot\left(\frac{2}{n}\right)^d\leq (2C)^{2-\alpha}<\infty$

for ${d=2-\alpha}$ . Since ${n\in\mathbb{N}}$ is arbitrary, we conclude the desired bound

$\displaystyle \textrm{dim}(G)\leq 2-\alpha$

A long-standing conjecture about the fractal geometry of ${W_{\lambda,b}}$ is:

Conjecture (Mandelbrot 1977): The Hausdorff dimension of the graph of ${W_{\lambda,b}}$ is

$\displaystyle 1<\textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b} < 2$

Remark 4 In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound

$\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) \geq 2 + \frac{\log\lambda}{\log b}$

Remark 5 The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).

In a recent paper (see here), Shen proved the following result:

Theorem 1 (Shen) For any ${b\geq 2}$ integer and for all ${1/b < \lambda < 1}$ , the Mandelbrot conjecture is true, i.e.,

$\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}$

Remark 6 The techniques employed by Shen also allow him to show that given ${\phi:\mathbb{R}\rightarrow\mathbb{R}}$ a ${\mathbb{Z}}$ -periodic, non-constant, ${C^2}$ function, and given ${b\geq 2}$ integer, there exists ${K=K(\phi,b)>1}$ such that

$\displaystyle \textrm{dim}(\textrm{graph}(f^{\phi}_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}$

for all ${1/K < \lambda < 1}$ .

Remark 7 A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all ${b\geq 2}$ integer, there exists ${1/b < \lambda_b < 1}$ such that

$\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}$

for all ${\lambda_b < \lambda < 1}$ .

The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem1 by discussing the particular case when ${1/b<\lambda<2/b}$ and ${b\in\mathbb{N}}$ is large.

2. Ledrappier’s dynamical approach

If ${b\geq 2}$ is an integer, then the self-similar function ${f^{\phi}_{\lambda,b}}$ (cf. Remark 1) is also ${\mathbb{Z}}$ -periodic, i.e., ${f^{\phi}_{\lambda,b}(x+1) = f^{\phi}_{\lambda,b}(x)}$ for all ${x\in\mathbb{R}}$ . In particular, if ${b\geq 2}$ is an integer, then ${\textrm{graph}(f^{\phi}_{\lambda,b})}$ is an invariant repeller for the endomorphism ${\Phi:\mathbb{R}/\mathbb{Z}\times\mathbb{R}\rightarrow \mathbb{R}/\mathbb{Z}\times\mathbb{R}}$ given by

$\displaystyle \Phi(x,y) = \left(bx\textrm{ mod }1, \frac{y-\phi(x)}{\lambda}\right)$

This dynamical characterization of ${G = \textrm{graph}(f^{\phi}_{\lambda,b})}$ led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when ${b\geq 2}$ is an integer.

Denote by ${\mathcal{A}}$ the alphabet ${\mathcal{A}=\{0,\dots,b-1\}}$ . The unstable manifolds of ${\Phi}$ through ${G}$ have slopes of the form

$\displaystyle (1,-\gamma \cdot s(x,u))$

where ${\frac{1}{b} < \gamma = \frac{1}{\lambda b} <1}$ , ${x\in\mathbb{R}}$ , ${u\in\mathcal{A}^{\mathbb{N}}}$ , and

$\displaystyle s(x,u):=\sum\limits_{n=0}^{\infty} \gamma^n \phi'\left(\frac{x + u_1 + u_2 b + \dots + u_n b^{n-1}}{b^n}\right)$

In this context, the push-forwards ${m_x := (u\mapsto s(x,u))_*\mathbb{P}}$ of the Bernoulli measure ${\mathbb{P}}$ on ${\mathcal{A}^{\mathbb{N}}}$ (induced by the discrete measure assigning weight ${1/b}$ to each letter of the alphabet ${\mathcal{A}}$ ) play the role of conditional measures along vertical fibers of the unique Sinai-Ruelle-Bowen (SRB) measure ${\theta}$ of the expanding endomorphism ${T:\mathbb{R}/\mathbb{Z}\times\mathbb{R} \rightarrow \mathbb{R}/\mathbb{Z}\times\mathbb{R}}$ ,

$\displaystyle T(x,y) = (bx\textrm{ mod }1, \gamma y + \psi(x)),$

where ${\gamma=1/\lambda b}$ and ${\psi(x)=\phi'(x)}$ . In plain terms, this means that

$\displaystyle \theta = \int_{\mathbb{R}/\mathbb{Z}} m_x \, d\textrm{Leb}(x) \ \ \ \ \ (1)$

where ${\theta}$ is the unique ${T}$ -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).

As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures ${m_x}$ have important consequences for the fractal geometry of the graph ${G}$ :

Theorem 2 (Ledrappier) Suppose that for Lebesgue almost every ${x\in\mathbb{R}}$ the conditional measures ${m_x}$ have dimension ${\textrm{dim}(m_x)=1}$ , i.e.,

$\displaystyle \lim\limits_{r\rightarrow 0}\frac{\log m_x(B(z,r))}{\log r} = 1 \textrm{ for } m_x\textrm{-a.e. } z$

Then, the graph ${G=\textrm{graph}(f^{\phi}_{\lambda,b})}$ has Hausdorff dimension

$\displaystyle \textrm{dim}(G) = 2 + \frac{\log\lambda}{\log b}$

Remark 8 Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that ${\textrm{dim}(G)\geq 2 + \frac{\log\lambda}{\log b}}$ . By Frostman lemma, we need to construct a Borel measure ${\nu}$ supported on ${G}$ such that

$\displaystyle \underline{\textrm{dim}}(\nu) := \textrm{ ess }\inf \underline{d}(\nu,x) \geq 2 + \frac{\log\lambda}{\log b}$

where ${\underline{d}(\nu,x):=\liminf\limits_{r\rightarrow 0}\log \nu(B(x,r))/\log r}$ . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure ${\mu^{\phi}_{\lambda, b}}$ given by the lift to ${G}$ of the Lebesgue measure on ${[0,1]}$ via the map ${x\mapsto (x,f^{\phi}_{\lambda,b}(x))}$ satisfies

$\displaystyle \underline{\textrm{dim}}(\mu^{\phi}_{\lambda,b}) \geq 2 + \frac{\log\lambda}{\log b}$

An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:

Corollary 3 If ${\theta}$ is absolutely continuous with respect to the Lebesgue measure ${\textrm{Leb}_{\mathbb{R}^2}}$ , then

$\displaystyle \textrm{dim}(G) = 2 + \frac{\log\lambda}{\log b}$

Proof: By (1), the absolute continuity of ${\theta}$ implies that ${m_x}$ is absolutely continuous with respect to ${\textrm{Leb}_{\mathbb{R}}}$ for Lebesgue almost every ${x\in\mathbb{R}}$ .

Since ${m_x\ll \textrm{Leb}_{\mathbb{R}}}$ for almost every ${x}$ implies that ${\textrm{dim}(m_x)=1}$ for almost every ${x}$ , the desired corollary now follows from Ledrappier’s theorem. $\Box$

3. Tsujii’s theorem

The relevance of Corollary 3 is explained by the fact that Tsujii found an explicittransversality condition implying the absolute continuity of ${\theta}$ .

More precisely, Tsujii firstly introduced the following definition:

Definition 4

Given ${\varepsilon>0}$ , ${\delta>0}$ and ${x_0\in\mathbb{R}/\mathbb{Z}}$ , we say that two infinite words ${u, v\in\mathcal{A}^{\mathbb{N}}}$ are ${(\varepsilon,\delta)}$ -transverse at ${x_0}$ if either
$\displaystyle |s(x_0,u)-s(x_0,v)|>\varepsilon$

or

$\displaystyle |s'(x_0,u)-s'(x_0,v)|>\delta$

Given ${q\in\mathbb{N}}$ , ${\varepsilon>0}$ , ${\delta>0}$ and ${x_0\in\mathbb{R}/\mathbb{Z}}$ , we say that two finite words ${k,l\in\mathcal{A}^q}$ are ${(\varepsilon,\delta)}$ -transverse at ${x_0}$ if ${ku}$ , ${lv}$ are ${(\varepsilon,\delta)}$ -transverse at ${x_0}$ for all pairs of infinite words ${u,v\in\mathcal{A}^{\mathbb{N}}}$ ; otherwise, we say that ${k}$ and ${l}$ are ${(\varepsilon,\delta)}$ -tangent at ${x_0}$ ;

${E(q,x_0;\varepsilon,\delta):= \{(k,l)\in\mathcal{A}^q\times\mathcal{A}^q: (k,l) \textrm{ is } (\varepsilon,\delta)\textrm{-tangent at } x_0\}}$

${E(q,x_0):=\bigcap\limits_{\varepsilon>0}\bigcap\limits_{\delta>0} E(q,x_0;\varepsilon,\delta)}$ ;

${e(q,x_0):=\max\limits_{k\in\mathcal{A}^q}\#\{l\in\mathcal{A}^q: (k,l)\in E(q,x_0)\}}$

${e(q):=\max\limits_{x_0\in\mathbb{R}/\mathbb{Z}} e(q,x_0)}$ .

Next, Tsujii proves the following result:

Theorem 5 (Tsujii) If there exists ${q\geq 1}$ integer such that ${e(q)<(\gamma b)^q}$ , then

$\displaystyle \theta\ll\textrm{Leb}_{\mathbb{R}^2}$

Remark 9 Intuitively, Tsujii’s theorem says the following. The transversality condition ${e(q)<(\gamma b)^q}$ implies that the majority of strong unstable manifolds ${\ell^{uu}}$ are mutually transverse, so that they almost fill a small neighborhood ${U}$ of some point ${x_0}$ (see the figure below extracted from this paper of Tsujii). Since the SRB measure ${\theta}$ is absolutely continuous along strong unstable manifolds, the fact that the ${\ell^{uu}}$ ‘s almost fill ${U}$ implies that ${\theta}$ becomes “comparable” to the restriction of the Lebesgue measure ${\textrm{Leb}_{\mathbb{R}^2}}$ to ${U}$ .

Remark 10 In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters ${\lambda}$ and ${b}$ , one has ${e(1)=1}$ . Indeed, once we know that ${e(1)=1}$ , since ${1<\gamma b}$ , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters ${\lambda}$ and ${b}$ .

For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:

Proposition 6 Let ${\phi(x) = \cos(2\pi x)}$ . If ${1/2<\gamma=1/\lambda b <1}$ and ${b\in\mathbb{N}}$ is sufficiently large, then

$\displaystyle e(1)<\gamma b$

In particular, by Corollary 3 and Tsujii’s theorem, if ${1/2<\gamma=1/\lambda b <1}$ and ${b\in\mathbb{N}}$ is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,

$\displaystyle \textrm{dim}(W_{\lambda,b}) = 2+\frac{\log\lambda}{\log b}$

Remark 11 The proof of Theorem 1 in full generality (i.e., for ${b\geq 2}$ integer and ${1/b<\lambda<1}$ ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function ${\sigma(q)\leq e(q)}$ (inspired from Peter-Paul inequality) and he proves

(a) a variant of Proposition 6: if ${b\geq 2}$ integer and ${1/b<\lambda<1}$ , then ${\sigma(q)<(\gamma b)^q}$ for some integer ${q}$ ;

(b) a variant of Tsujii’s theorem: if ${\sigma(q)<(\gamma b)^q}$ for some integer ${q}$ , then ${\theta\ll\textrm{Leb}_{\mathbb{R}^2}}$ .

See Sections 2, 3, 4 and 5 of Shen’s paper for more details.

We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by

$\displaystyle s(x,u):=-2\pi\sum\limits_{n=0}^{\infty} \gamma^n \sin\left(2\pi\frac{x + u_1 + u_2 b + \dots + u_n b^{n-1}}{b^n}\right)$

for ${x\in\mathbb{R}}$ , ${u\in\mathcal{A}^{\mathbb{N}}}$ , so that

$\displaystyle s'(x,u)=-4\pi^2\sum\limits_{n=0}^{\infty} \left(\frac{\gamma}{b}\right)^n \cos\left(2\pi\frac{x + u_1 + u_2 b + \dots + u_n b^{n-1}}{b^n}\right)$

Remark 12 Since ${\gamma/b < \gamma}$ , the series defining ${s'(x,u)}$ converges faster than the series defining ${s(x,u)}$ .

By studying the first term of the expansion of ${s(x,u)}$ and ${s'(x,u)}$ (while treating the remaining terms as a “small error term”), it is possible to show that if ${(k,l)\in E(1,x_0)}$ , then

$\displaystyle \left|\sin\left(2\pi\frac{x_0+k}{b}\right) - \sin\left(2\pi\frac{x_0+l}{b}\right)\right| \leq\frac{2\gamma}{1-\gamma} \ \ \ \ \ (2)$

and

$\displaystyle \left|\cos\left(2\pi\frac{x_0+k}{b}\right) - \cos\left(2\pi\frac{x_0+l}{b}\right)\right| \leq \frac{2\gamma}{b-\gamma} \ \ \ \ \ (3)$

(cf. Lemma 3.2 in Shen’s paper).

Using these estimates, we can find an upper bound for ${e(1)}$ as follows. Take ${x_0\in\mathbb{R}/\mathbb{Z}}$ with ${e(1)=e(1,x_0)}$ , and let ${k\in\mathcal{A}}$ be such that ${(k,l_1),\dots,(k,l_{e(1)})\in E(1,x_0)}$ distinct elements listed in such a way that

$\displaystyle \sin(2\pi x_i)\leq \sin(2\pi x_{i+1})$

for all ${i=1,\dots,e(1)-1}$ , where ${x_i:=(x_0+l_i)/b}$ .

From (3), we see that

$\displaystyle \left|\cos\left(2\pi x_i\right) - \cos\left(2\pi x_{i+1}\right)\right| \leq \frac{4\gamma}{b-\gamma}$

for all ${i=1,\dots,e(1)-1}$ .

Since

$\displaystyle (\cos(2\pi x_i)-\cos(2\pi x_{i+1}))^2 + (\sin(2\pi x_i)-\sin(2\pi x_{i+1}))^2 = 4\sin^2(\pi(x_i-x_{i+1}))\geq 4\sin^2(\pi/b),$

it follows that

$\displaystyle |\sin(2\pi x_i)-\sin(2\pi x_{i+1})|\geq \sqrt{4\sin^2\left(\frac{\pi}{b}\right) - \left(\frac{4\gamma}{b-\gamma}\right)^2} \ \ \ \ \ (4)$

Now, we observe that

$\displaystyle \sqrt{4\sin^2\left(\frac{\pi}{b}\right) - \left(\frac{4\gamma}{b-\gamma}\right)^2} > \frac{4}{b} \ \ \ \ \ (5)$

for ${b}$ large enough. Indeed, this happens because

${\sqrt{z^2-w^2}>2(z-w)}$ if ${z+w>4(z-w)}$ ;
${z+w>4(z-w)}$ if ${z/w:=u < 5/3}$ ;
${\frac{2\sin(\frac{\pi}{b})}{\frac{4\gamma}{b-\gamma}}\rightarrow \frac{2\pi}{4\gamma} (< \frac{5}{3})}$ as ${b\rightarrow\infty}$ , and ${2\sin(\frac{\pi}{b}) - \frac{4\gamma}{b-\gamma} \rightarrow (2\pi-4\gamma)\frac{1}{b} (>\frac{2}{b})}$ as ${b\rightarrow\infty}$ (here we used ${\gamma<1}$ ).

By combining (4) and (5), we deduce that

$\displaystyle |\sin(2\pi x_i)-\sin(2\pi x_{i+1})| > 4/b$

for all ${i=1,\dots, e(1)-1}$ .

Since ${-1\leq\sin(2\pi x_1)\leq\sin(2\pi x_2)\leq\dots\leq\sin(2\pi x_{e(1)})\leq 1}$ , the previous estimate implies that

$\displaystyle \frac{4}{b}(e(1)-1)<\sum\limits_{i=1}^{e(1)-1}(\sin(2\pi x_{i+1}) - \sin(2\pi x_i)) = \sin(2\pi x_{e(1)}) - \sin(2\pi x_1)\leq 2,$

i.e.,

$\displaystyle e(1)<1+\frac{b}{2}$

Thus, it follows from our assumptions ( ${\gamma>1/2}$ , ${b}$ large) that

$\displaystyle e(1)<1+\frac{b}{2}<\gamma b$

This completes the (sketch of) proof of Proposition 6 (and our discussion of Shen’s talk).

One Dimensional Dynamical System

低维动力系统

January 27, 2016 zr9558 Leave a comment

One Dimensional Real and Complex Dynamics需要学习的资料：

复分析基础：本科生课程

(1) Complex Analysis, 3rd Edition, Lars V. Ahlfors

(2) Complex Analysis, Elias M. Stein

进阶复分析：研究生课程

(1) Lectures on Riemann Surfaces (GTM 81), Otto Forster

(2) Lectures on Quasiconformal Mappings, Lars V. Ahlfors

实分析基础：本科生课程

(1) Real Analysis, Rudin

(2) Real Analysis, Elias M. Stein

专业书籍：

实动力系统：

(1) One Dimensional Dynamics, Welington de Melo & Sebastian VanStrien

(2) Mathematical Tools for One-Dimensional Dynamics (Cambridge Studies in Advanced Mathematics), Edson de Faria / Welington de Melo

复动力系统：

(3) Dynamics in One Complex Variable, John Milnor

(4) Complex Dynamics, Lennart Carleson

(5) Complex Dynamics and Renormalization, Curtis T. McMullen

(6) Renormalization and 3-Manifolds Which Fiber over the Circle, Curtis T. McMullen

(7) Iteration of rational functions (GTM 132), Alan F. Beardon

遍历论：

(8) An Introduction to Ergodic Theory (GTM 79), Walters Peter

One Dimensional Dynamical System

Ergodic Properties

December 5, 2014 zr9558 Leave a comment

One Dimensional Dynamics

— Welington De Melo, Sebastian van Strien

Chapter 5. Ergodic Properties and Invariant Measures.

1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.

A distortion result for unimodal maps with recurrence

Given a unimodal map $f$ , we say that an interval $U$ is symmetric if $\tau(U)=U$ where $\tau:[-1,1]\rightarrow [-1,1]$ is so that $f(\tau(x))=f(x)$ and $\tau(x)\neq x$ if $x\neq c$ . Furthermore, for each symmetric interval $U$ let

$D_{U}=\{x: \text{ there exists } k>0 \text{ with } f^{k}(x)\in U\};$

for $x\in D_{U}$ let $k(x,U)$ be the minimal positive integer with $f^{k}(x)\in U$ and let

$R_{U}(x)=f^{k(x,U)}(x).$

We call $R_{U}: D_{U}\rightarrow U$ the Poincare map or transfer map to $U$ and $k(x,U)$ the transfer time of $x$ to $U$ . The distortion result states that one can fined a sequence of symmetric neighbourhoods of the turning point such that the Poincare maps to these intervals have a distortion which is universally bounded:

Theorem 1.1. Let $f:[-1,1]\rightarrow [-1,1]$ be a unimodal map with one non-flat critical point with negative Schwarzian derivative and without attracting periodic points. Then there exists $\rho>0$ and a sequence os symmetric intervals $U_{n}\subseteq V_{n}$ around the turning point which shrink to $c$ such that $V_{n}$ contains a $\rho-$ scaled neighbourhood of $U_{n}$ and such that the following properties hold.

1. The transfer time on each component of $D_{U_{n}}$ is constant.

2. Let $I_{n}$ be a component of the domain $D_{U_{n}}$ of the transfer map to $U_{n}$ which does not intersect $U_{n}$ . Then there exists an interval $T_{n}\supseteq I_{n}$ such that $f^{k}|T_{n}$ is monotone, $f^{k}(T_{n})\supseteq V_{n}$ and $f^{k}(I_{n})=U_{n}$ . Here $k$ is the transfer time on $I_{n}$ , i.e., $R_{U_{n}}|I_{n}=f^{k}$ .

Corollary. There exists $K<\infty$ such that

1. for each component $I_{n}$ of $D_{U_{n}}$ not intersecting $U_{n}$ , the transfer map $R_{U_{n}}$ to $U_{n}$ sends $I_{n}$ diffeomorphically onto $U_{n}$ and the distortion of $R_{U_{n}}$ on $I_{n}$ is bounded from above by $K$ .

2. on each component $I_{n}$ of $D_{U_{n}}$ which is contained in $U_{n}$ , the map $R_{U_{n}}:I_{n}\rightarrow U_{n}$ can be written as $(f^{k(n)-1}|f(I_{n}))\circ f|I_{n}$ where the distortion of $f^{k(n)}|f(I_{n})$ is universally bounded by $K$ .

As before, we say that $f$ is ergodic with respect to the Lebesgue measure if each completely invariant set $X$ (Here $X$ is called completely invariant if $f^{-1}(X)=X$ ) has either zero or full Lebesgue measure. An alternative way to define this notation of ergodicity goes as follows: $f$ is ergodic if for each two forward invariant sets $X$ and $Y$ such that $X\cap Y$ has Lebesgue measure zero, at most one of these sets has positive Lebesgue measure. (Here $X$ is called forward invariant if $f(X)\subseteq X$ .)

Theorem 1.2 (Blokh and Lyubich). Let $f:[-1,1]\rightarrow [-1,1]$ be a unimodal map with a non-flat critical point with negative Schwarzian derivative and without an attracting periodic points. Then $f$ is ergodic with respect to the Lebesgue measure.

Theorem 1.3. Let $f:[-1,1]\rightarrow [-1,1]$ be a unimodal map with a non-flat critical point with negative Schwarzian derivative. Then $f$ has a unique attractor $A$ , $\omega(x)=A$ for almost all $x$ and $A$ either consists of intervals or has Lebesgue measure zero. Furthermore, one has the following:

1. if $f$ has an attracting periodic orbit then $A$ is this periodic orbit;

2. if $f$ is infinitely often renormalizable then $A$ is the attracting Cantor set $\omega(c)$ (in which case it is called a solenoidal attractor);

3. $f$ is only finitely often renormalizable then either

(a) $A$ coincides with the union of the transitive intervals, or,

(b) $A$ is a Cantor set and equal to $\omega(c)$ .

If $\omega(c)$ is not a minimal set then $f$ is as in case 3.a and each closed forward invariant set either contains intervals or has Lebesgue measure zero. Moreover, if $\omega(c)$ does not contain intervals, then $\omega(c)$ has Lebesgue measure zero.

Remark. Here a forward invariant set $X$ is said to be minimal if the closure of the forward orbit of a point in $X$ is always equal to $X$ . The attractors in case 3.b is called a non-renormalizable attracting Cantor set, or absorbing Cantor attractor or wild Cantor attractor. Such an attractor really exists which is proven in [BKNS], and one has the following strange phenomenon: there exist many orbits which are dense in some finite union of intervals and yet almost all points tend to a minimal Cantor set of Lebesgue measure zero (this Cantor set is $\omega(c)$ ). The Fibonacci map is non-renormalizable and for which $\omega(c)$ is a Cantor set. It was shown by Lyubich and Milnor that the quadratic map with this dynamics has no absorbing Cantor attractors. More generally, Jakobson and Swiatek proved that maps with negative Schwarzian derivative and which are close to the map $f(x)=4x(1-x)$ do not have such Cantor attractors. Moreover, Lyubich has shown that these absorbing Cantor attractors can not exist if the critical point is quadratic. However, Bruin, Keller, Nowicki and Van Strien showed that the absorbing Cantor attractors exist for Fibonacci maps when the critical order $\ell$ is sufficiently large enough.

Theorem (Lyubich). If $f:[-1,1]\rightarrow [-1,1]$ is $C^{3}$ unimodal, has a quadratic critical point, has negative Schwarzian derivative and has no periodic attractors, then each closed forward invariant set $K$ which has positive Lebesgue measure contains an interval.

The next result, which is due to Martens (1990), shows that if these absorbing Cantor attractors do not exist then one has a lot of ‘expansion’. Let $x$ not be in the pre orbit of $c$ and define $T_{n}(x)$ to be the maximal interval on which $f^{n}|T_{n}(x)$ is monotone. Let $R_{n}(x)$ and $L_{n}(x)$ be the components of $T_{n}\setminus x$ and define $r_{n}(x)$ be the minimum of the length of $f^{n}(R_{n}(x))$ and $f^{n}(L_{n}(x))$ .

Theorem 1.4 (Martens). Let $f$ be a $C^{3}$ unimodal map with negative Schwarian derivative whose critical point is non-flat. Then the following three properties are equivalent.

1. $f$ has no absorbing Cantor attractor;

2. $\limsup_{n\rightarrow \infty} r_{n}(x)>0$ for almost all $x$ ;

3. there exist neighbourhoods $U\subseteq V$ of $c$ with $cl(U)\subseteq int(V)$ such that for almost every $x$ there exists a positive integer $m$ and an interval neighbourhood $T$ of $x$ such that $f^{m}|T$ is monotone, $f^{m}(T)\supseteq V$ and $f^{m}(x)\in U$ .

One Dimensional Dynamical System

Perron-Frobenius Operator

November 5, 2014 zr9558 Leave a comment

Perron-Frobenius Operator

Consider a map $f$ which possibly has a finite (or countable) number of discontinuities or points where possibly the derivative does not exist. We assume that there are points

$\displaystyle q_{0}<q_{1}<\cdot\cdot\cdot <q_{k}$ or $q_{0}<q_{1}<\cdot\cdot\cdot<q_{\infty}<\infty$

such that $f$ restricted to each open interval $A_{j}=(q_{j-1},q_{j})$ is $C^{2}$ , with a bound on the first and the second derivatives. Assume that the interval $[q_{0},q_{k}]$ ( or $[q_{0},q_{\infty}]$ ) is positive invariant, so $f(x)\in [q_{0},q_{k}]$ for all $x\in [q_{0}, q_{k}]$ ( or $f(x)\in [q_{0},q_{\infty}]$ for all $x\in[q_{0},q_{\infty}]$ ).

For such a map, we want a construction of a sequence of density functions that converge to a density function of an invariant measure. Starting with $\rho_{0}(x)\equiv(q_{k}-q_{0})^{-1}$ ( or $\rho_{0}(x)\equiv(q_{\infty}-q_{0})^{-1}$ ),assume that we have defined densities up to $\rho_{n}(x)$ , then define define $\rho_{n+1}(x)$ as follows

$\displaystyle \rho_{n+1}(x)=P(\rho_{n})(x)=\sum_{y\in f^{-1}(x)}\frac{\rho_{n}(y)}{|Df(y)|}.$

This operator $P$ , which takes one density function to another function, is called the Perron-Frobenius operator. The limit of the first $n$ density functions converges to a density function $\rho^{*}(x)$ ,

$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x).$

The construction guarantees that $\rho^{*}(x)$ is the density function for an invariant measure $\mu_{\rho^{*}}$ .

Example 1. Let

$\displaystyle f(x)= \begin{cases} x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2x &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

We construct the first few density functions by applying the Perron-Frobenius operator, which indicates the form of the invariant density function.
Take $\rho_{0}(x)\equiv1$ on $[0,1]$ . From the definition of $f(x)$ , the slope on $(0,\frac{1}{2})$ and $(\frac{1}{2},1)$ are 1 and 2, respectively. If $x\in (\frac{1}{2},1)$ , then it has only one pre-image on $(\frac{1}{2},1)$ ; else if $x\in(0,\frac{1}{2})$ , then it has two pre-images, one is $x^{'}$ in $(0,\frac{1}{2})$ , the other one is $x^{''}$ in $(\frac{1}{2},1)$ . Therefore,

$\rho_{1}(x)= \begin{cases} \frac{1}{1}+\frac{1}{2} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{1}{2} &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

By similar considerations,

$\displaystyle \rho_{2}(x)=\begin{cases}1+\frac{1}{2}+\frac{1}{2^{2}} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{1}{2^{2}} &\mbox{if } x\in(\frac{1}{2},1).\end{cases}$

By induction, we get

$\displaystyle \rho_{n}(x)=\begin{cases}1+\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{2^{n}} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{1}{2^{n}} &\mbox{if } x\in(\frac{1}{2},1).\end{cases}$

Now, we begin to calculate the density function $\rho^{*}(x)$ . If $x\in(0,\frac{1}{2})$ , then
$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x) =\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1} \sum_{m=0}^{n}\frac{1}{2^{m}} =\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\left(2-\frac{1}{2^{n}}\right)=2.$
If $x\in(\frac{1}{2},1)$ , then
$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x) =\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\frac{1}{2^{n}} =\lim_{k\rightarrow \infty}\frac{1}{k}\left(2-\frac{1}{2^{k}}\right)=0.$
i.e.

$\displaystyle \rho^{*}(x)= \begin{cases} 2 &\mbox{if } x\in(0,\frac{1}{2}), \\ 0 &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

Example 2. Let

$\displaystyle f(x)=\begin{cases} 2x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2x-1 &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

Take $\rho_{0}(x)\equiv1$ on $(0,1)$ . By induction, $\rho_{n}(x)\equiv1$ on $(0,1)$ for all $n\geq 0$ . Therefore, $\rho^{*}(x)\equiv1$ on $(0,1)$ .

Example 3. Let

$\displaystyle f(x)=\begin{cases} x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2^{n+1}\cdot\left(x-\left(1-\frac{1}{2^{n}}\right)\right) &\mbox{if } x\in\left(1-\frac{1}{2^{n}},1-\frac{1}{2^{n+1}}\right) \text{ for all } n\geq 1.\end{cases}$

Take $\rho_{0}(x)\equiv1$ on $(0,1)$ . Assume

$\displaystyle \rho_{n}(x)= \begin{cases} a_{n} &\mbox{if } x\in(0,\frac{1}{2}), \\ b_{n} &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

for all $n\geq 0$ . It is obviously that $a_{0}=b_{0}=1$ . By similar considerations,
$\displaystyle \rho_{n+1}(x)= \begin{cases} \frac{a_{n}}{1}+\frac{b_{n}}{4}+\frac{b_{n}}{8}+\frac{b_{n}}{16}+\cdot\cdot\cdot= a_{n}+\frac{b_{n}}{2} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{b_{n}}{4}+\frac{b_{n}}{8}+\frac{b_{n}}{16}+\cdot\cdot\cdot = \frac{b_{n}}{2} &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$
That means

$\displaystyle \left( \begin{array}{ccc} a_{n+1} \\ b_{n+1} \end{array} \right) =\left( \begin{array}{ccc} a_{n}+\frac{1}{2}b_{n} \\ \frac{1}{2}b_{n} \end{array} \right) = \left( \begin{array}{ccc} 1 & \frac{1}{2} \\ 0 & 1 \end{array} \right) \left( \begin{array}{ccc} a_{n} \\ b_{n} \end{array} \right)$

for all $n\geq 0$ . From direct calculation, $\displaystyle a_{n}=2-\frac{1}{2^{n}}$ and $\displaystyle b_{n}=\frac{1}{2^{n}}$ for all $n\geq 0$ . Therefore,

$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x)=\begin{cases} 2 &\mbox{if } x\in (0,\frac{1}{2}), \\ 0 &\mbox{if } x\in (\frac{1}{2},1). \end{cases}$

Example 4. Let

$\displaystyle f(x)=\begin{cases} 1.5 x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2^{n+1}\cdot\left(x-\left(1-\frac{1}{2^{n}}\right)\right) &\mbox{if } x\in\left(1-\frac{1}{2^{n}},1-\frac{1}{2^{n+1}}\right) \text{ for all } n\geq 1.\end{cases}$

Take $\rho_{0}(x)\equiv1$ on $(0,1)$ . Assume

$\displaystyle \rho_{n}(x)= \begin{cases} a_{n} &\mbox{if } x\in(0,\frac{3}{4}), \\ b_{n} &\mbox{if } x\in(\frac{3}{4},1). \end{cases}$

for all $n\geq 0$ . It is obviously that $a_{0}=b_{0}=1$ . By similar considerations,

$\displaystyle \left( \begin{array}{ccc} a_{n+1} \\ b_{n+1} \end{array} \right) =\left( \begin{array}{ccc} \frac{11}{12}a_{n}+\frac{1}{4}b_{n} \\ \frac{1}{4}a_{n}+\frac{1}{4}b_{n} \end{array} \right) = \left( \begin{array}{ccc} \frac{11}{12} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} \end{array} \right) \left( \begin{array}{ccc} a_{n} \\ b_{n} \end{array} \right)$

for all $n\geq 0$ . From matrix diagonalization , $\displaystyle a_{n}=\frac{6}{5}-\frac{1}{5}\cdot\frac{1}{6^{n}}$ and $\displaystyle b_{n}=\frac{2}{5}+\frac{3}{5}\cdot\frac{1}{6^{n}}$ for all $n\geq 0$ .

Therefore,

$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x)=\begin{cases} \frac{6}{5} &\mbox{if } x\in (0,\frac{3}{4}), \\ \frac{2}{5} &\mbox{if } x\in (\frac{3}{4},1). \end{cases}$

Perron-Frobenius Theory

Definition. Let $A=[a_{ij}]$ be a $k\times k$ matrix. We say $A$ is non-negative if $a_{ij}\geq 0$ for all $i,j$ . Such a matrix is called irreducible if for any pair $i,j$ there exists some $n>0$ such that $a_{ij}^{(n)}>0$ where $a_{ij}^{(n)}$ is the $(i,j)-$ th element of $A^{n}$ . The matrix $A$ is irreducible and aperiodic if there exists $n>0$ such that $a_{ij}^{(n)}>0$ for all $i,j$ .

Perron-Frobenius Theorem Let $A=[a_{ij}]$ be a non-negative $k\times k$ matrix.

(i) There is a non-negative eigenvalue $\lambda$ such that no eigenvalue of $A$ has absolute value greater than $\lambda$ .

(ii) We have $\min_{i}(\sum_{j=1}^{k}a_{ij})\leq \lambda\leq \max_{i}(\sum_{j=1}^{k}a_{ij})$ .

(iii) Corresponding to the eigenvalue $\lambda$ there is a non-negative left (row) eigenvector $u=(u_{1},\cdot\cdot\cdot, u_{k})$ and a non-negative right (column) eigenvector $v=(v_{1},\cdot\cdot\cdot, v_{k})^{T}$ .

(iv) If $A$ is irreducible then $\lambda$ is a simple eigenvalue and the corresponding eigenvectors are strictly positive (i.e. $u_{i}>0$ , $v_{i}>0$ all $i$ ).

(v) If $A$ is irreducible then $\lambda$ is the only eigenvalue of $A$ with a non-negative eigenvector.

Theorem.
Let $A$ be an irreducible and aperiodic non-negative matrix. Let $u=(u_{1},\cdot\cdot\cdot, u_{k})$ and $v=(v_{1},\cdot\cdot\cdot, v_{k})^{T}$ be the strictly positive eigenvectors corresponding to the largest eigenvalue $\lambda$ as in the previous theorem. Then for each pair $i,j$ , $\lim_{n\rightarrow \infty} \lambda^{-n}a_{ij}^{(n)}=u_{j}v_{i}$ .

Now, let us see previous examples, again. The matrix $A$ is irreducible and aperiodic non-negative matrix, and $\lambda=1$ has the largest absolute value in the set of all eigenvalues of $A$ . From Perron-Frobenius Theorem, $u_{i}, v_{j}>0$ for all pairs $i,j$ . Then for each pari $i,j$ ,
$\lim_{n\rightarrow \infty}a_{ij}^{(n)}=u_{j}v_{i}$ . That means $\lim_{n\rightarrow \infty}A^{(n)}$ is a strictly positive $k\times k$ matrix.

Markov Maps

Definition of Markov Maps. Let $N$ be a compact interval. A $C^{1}$ map $f:N\rightarrow N$ is called Markov if there exists a finite or countable family $I_{i}$ of disjoint open intervals in $N$ such that

(a) $N\setminus \cup_{i}I_{i}$ has Lebesgue measure zero and there exist $C>0$ and $\gamma>0$ such that for each $n\in \mathbb{N}$ and each interval $I$ such that $f^{j}(I)$ is contained in one of the intervals $I_{i}$ for each $j=0,1,...,n$ one has

$\displaystyle \left| \frac{Df^{n}(x)}{Df^{n}(y)}-1 \right| \leq C\cdot |f^{n}(x)-f^{n}(y)|^{\gamma} \text{ for all } x,y\in I;$

(b) if $f(I_{k})\cap I_{j}\neq \emptyset$ , then $f(I_{k})\supseteq I_{j}$ ;

As usual, let $\lambda$ be the Lebesgue measure on $N$ . We may assume that $\lambda$ is a probability measure, i.e., $\lambda(N)=1$ . Usually, we will denote the Lebesgue measure of a Borel set $A$ by $|A|$ .

Theorem. Let $f:N\rightarrow N$ be a Markov map and let $\cup_{i}I_{i}$ be corresponding partition. Then there exists a $f-$ invariant probability measure $\mu$ on the Borel sets of $N$ which is absolutely continuous with respect to the Lebesgue measure $\lambda$ . This measure satisfies the following properties:

(a) its density $\frac{d\mu}{d\lambda}$ is uniformly bounded and Holder continuous. Moreover, for each $i$ the density is either zero on $I_{i}$ or uniformly bounded away from zero.

If for every $i$ and $j$ one has $f^{n}(I_{j})\supseteq I_{i}$ for some $n\geq 1$ then

(b) the measure is unique and its density $\frac{d\mu}{d\lambda}$ is strictly positive;

(d) $\lim_{n\rightarrow \infty} |f^{-n}(A)|=\mu(A)$ for every Borel set $A\subseteq N$ .

If $f(I_{i})=N$ for each interval $I_{i}$ , then

(e) the density of $\mu$ is also uniformly bounded from below.

Graduate Summer School 2008

2. 刚性定理

July 18, 2014 zr9558 Leave a comment

考虑二次多项式 $f_{a}(x)=ax(1-x), a\in[0,4]$ , $f_{a}:[0,1]\rightarrow [0,1]$ .

问题：

$\{ a\in[0,4]: f_{a} \text{ satisfies Axiom A} \}$ 是否在 [0,4] 中稠密？

引理： $f_{a}$ 满足 Axiom A $\Leftrightarrow$ $f_{a}$ 有双曲吸引周期轨。

定义 Kneading 序列： $K(f_{a})=\{ i_{1}, i_{2},... \},$ $i_{k}=L \text{ if } f_{a}^{k}(\frac{1}{2})<\frac{1}{2};$ $i_{k}=c=\frac{1}{2} \text{ if } f_{a}^{k}(\frac{1}{2})=\frac{1}{2};$ $i_{k}=R \text{ if } f_{a}^{k}(\frac{1}{2})>\frac{1}{2}.$

例子：

$K(f_{4})=(R,L,L,L,...)=RLLL,$

$K(f_{1})=(L,L,L,L,...)=LLLL,$

$K(f_{2})=(c,c,c,c,...)=cccc,$

$K(f_{1.9})=(L,L,L,L,...)=LLLL,$

在这里， $f_{1}$ 和 $f_{1.9}$ 不是拓扑共轭的，即使它们的 Kneading 序列是一样的。

定义：f 和 g 称为拓扑共轭，如果存在同胚映射 h 使得 $h\circ f= g \circ h$ .

性质1: 如果 $f_{a_{1}}$ 和 $f_{a_{2}}$ 拓扑同胚，则有 $K(f_{a_{1}})= K(f_{a_{2}}).$

引理：如果 $f_{a_{1}}$ 和 $f_{a_{2}}$ 没有双曲吸引或者双曲中性周期轨，则 $K(f_{a_{1}})= K(f_{a_{2}}) \Rightarrow$ $f_{a_{1}}$ 和 $f_{a_{2}}$ 拓扑同胚 $\Rightarrow a_{1}=a_{2}.$

定义：拟共形映射的分析定义： $\varphi: \Omega\rightarrow \tilde{\Omega},$ 在这里 $\Omega, \tilde{\Omega}$ 都是复平面上面的连通开集, $\varphi$ 是保持定向的同胚映射，称 $\varphi$ 是 K 拟共形映射 ( $K\geq 1$ ), 如果

(1) $\varphi$ 是 ACL 的，也就是线段上绝对连续，absolutely continuous on lines.

(2) $| \frac{\partial \varphi}{\partial \overline{z}} | \leq \frac{K-1}{K+1} |\frac{\partial \varphi}{\partial z}|$ 几乎处处成立。

拟共形映射的一些性质：假设 $\varphi$ 是 K－拟共形映射， $K\geq 1$ .

(i) $\varphi$ 几乎处处可微。对几乎所有的 $z_{0}\in \Omega$ 有

$\varphi(z) = \varphi(z_{0}) + \frac{\partial \varphi}{\partial z}(z_{0})(z-z_{0}) + \frac{\partial \varphi}{\partial \overline{z}}(z_{0})\overline{(z-z_{0})}+ o(|z-z_{0}|).$

$| \frac{\partial \varphi}{\partial z}|>0$ 几乎处处成立。

定义 $\varphi$ 的复特征是 $\mu_{\varphi}= \frac{\partial \varphi}{\partial \overline{z}} / \frac{\partial \varphi}{\partial z},$ 则 $||\mu_{\varphi}||_{\infty} \leq \frac{K-1}{K+1} <1.$

(ii) Measurable Riemann Mapping Theorem ( Ahlfors-Bers )

Assume $f_{a}(x)=ax(1-x),$ $a_{0} \in (0,4]$

$Comb(a_{0})=\{ a\in(0,4]: K(f_{a})=K(f_{a_{0}}) \},$

$Top(a_{0})= \{ a\in (0,4]: f_{a} \text{ and } f_{a_{0}} \text{ are topological conjugate } \},$

$\Rightarrow Top(a_{0}) \subseteq Comb(a_{0}).$

$Qc(a_{0}) = \{ a\in (0,4]: f_{a} \text{ and } f_{a_{0}} \text{ are quasi-conformal conjugate} \},$

$Aff(a_{0}) = \{ a\in (0,4]: f_{a} \text{ and } f_{a_{0}} \text{ are linear conjugate} \},$

$\Rightarrow Aff(a_{0}) \subseteq Qc(a_{0}).$

刚性问题： $Comb(a_{0})=Qc(a_{0}) ?$ $Comb(a_{0})=Aff(a_{0})?$

定理：( Graczyk – Swiatek, Lyubich, 1997) 假设 $f_{a_{0}}$ 没有双曲吸引或者中性周期轨，则 $Comb(a_{0})=Qc(a_{0}).$

推论：( Sullivan, 1988) Axiom A 系统在实系数二次多项式中稠密。

Graduate Summer School 2008

1.一维动力系统中的双曲性

July 18, 2014 zr9558 2 Comments

定理： 假设 $f:[0,1]\rightarrow [0,1]$ 是 $C^{k}$ , $k$ 是正整数，则存在 $C^{k}$ 函数 $f_{n}:[0,1]\rightarrow [0,1]$ 使得 $|| f_{n}- f ||_{C^{k}}=\max_{x\in[0,1]} \max_{0\leq m\leq k} |D^{m}f_{n}(x)-D^{m}f(x)| \rightarrow 0$ as $k\rightarrow \infty$ , 这里的每个 $f_{n}$ 都满足Axiom A。

假设 $X$ 是紧致度量空间， $f:X\rightarrow X$ 是连续函数。如果 $n$ 是使得 $f^{n}(x)=x$ 的最小正整数，则称x是以n为周期的周期点。

定义：

$\omega(x)=\{ y\in X: \exists n_{k} \rightarrow \infty, f^{n_{k}}(x)\rightarrow y\}$ .

正向不变集： $f(A)\subseteq A$ ,

反向不变集： $f^{-1}(A)\subseteq A$ ,

完全不变集： $f^{-1}(A)=A$ i.e. $f(A)\subseteq A$ and $f^{-1}(A)\subseteq A$ .

假设 $X=[0,1]$ , $f(x)\in C^{1}[0,1]$ , $\{ x, f(x), ... , f^{n-1}(x)\}$ 是以n为周期的周期轨道，定义乘子(multiplier) $\lambda=Df^{n}(x)=Df(x)\cdot Df(f(x)) ... Df(f^{n-1}(x))$ 。

$|\lambda| \neq 1$ 称为 $orb(x)=\{ f^{k}(x): k=0,1,2... \}$ 是双曲周期轨。

$|\lambda|=1$ 称为中性周期轨。

$|\lambda|<1$ 称为双曲吸引轨。

$|\lambda|>1$ 称为双曲斥性轨。

双曲集合（hyperbolic set）:假设 $f:[0,1]\rightarrow [0,1]$ 是 $C^{1}$ 映射，A是紧集并且 $f(A)\subseteq A$ 。如果存在 $C>0, \lambda>1$ 使得对任意的 $x\in A, n\geq 1$ , 有 $|Df^{n}(x)| \geq C\lambda^{n}$ ，则称A是双曲集。

Axiom A：假设 $f:[0,1]\rightarrow [0,1]$ 是 $C^{1}$ 映射，称 f 满足 Axiom A是指：

（1）f 有有限多个双曲吸引轨 $\theta_{1},...,\theta_{m}$ ,

（2） $B(\theta_{i})$ 是双曲吸引轨 $\theta_{i}$ 的吸引区域, $\Omega=[0,1]\setminus \cup_{i=1}^{m}B(\theta_{i})$ 是双曲集。

例子1： $f(x)=-x^{2}$ ，1是双曲斥性不动点，0是双曲吸引不动点。 $B(\{0\})=(-1,1)$ , $\Omega=[-1,1]\setminus B(\{0\})=\{-1,1\}$ . $f^{n}(x)=x^{2^{n}}$ , $Df^{n}(x)=2^{n}x^{2^{n}-1}$ . 取 $C=1, \lambda=2$ .

例子2： $f(x)=2x(1-x), f(x)=ax(1-x)$ .

性质1: 双曲斥性周期轨一定是双曲集。

性质2: 双曲集中没有临界点。

性质3: 双曲集合中任何一个周期轨都是双曲斥性的。

命题：假设 $f:[0,1]\rightarrow [0,1]$ 属于 $C^{1+\alpha}$ 并且 $\alpha \in(0,1)$ . i.e. $Df(x)$ 是 $\alpha-Holder$ 连续的， $|Df(x)-Df(y)|\leq C|x-y|^{\alpha}.$ 如果A是双曲集，则A的Lebesgue测度是零。

证明：

定理（Mane，1985）（CMP）
假设 $f:[0,1]\rightarrow [0,1]$ 是一个 $C^{2}$ 的映射，

(1) f 的所有周期轨都是双曲的。

(2) Crit(f) 指的是 f 的临界点。 $\forall c\in Crit(f)$ , 则存在双曲吸引周期轨 $\theta_{c}$ 使得 $d(f^{n}(c),\theta_{c})\rightarrow 0, n\rightarrow \infty.$

$\Longleftrightarrow$ f 满足 Axiom A。

另外一种形式：

假设 $f:[0,1]\rightarrow [0,1]$ 是一个 $C^{2}$ 的映射，

$U\subseteq Crit(f)\cup \text{ hyperbolic attracting orbits }\cup \text{ and neutral orbits }$ ,

$\Lambda_{U} = \{ x\in[0,1]: f^{n}(x)\notin U, \forall n\geq 0 \}$ ,

$\Rightarrow$ $\Lambda_{U}$ 是双曲集。

One Dimensional Dynamical System

The Cross Ratio Tool and the Koebe Principle

October 14, 2013 zr9558 Leave a comment

Let $j \subseteq t$ be intervals and let l, r be the components of $t \setminus j$ . Then the Cross Ratio is defined as

$C(t,j) = (|t| \cdot |j|) / ( |l| \cdot |r|).$

Assume g is a $C^{3}$ monotone function on the interval t, and g(t)=T, g(j)=J, g(l)=L, g(r)=R. Then define

$B(g,t,j)=\frac{C(T,J)}{C(t,j)} = \frac{|T|\cdot |J|}{|L| \cdot |R|} \cdot \frac{|l|\cdot |r|}{|t|\cdot |j|}.$

Define the Schwarzian Derivative for $C^{3}$ function g,

$Sg(x)=\frac{D^{3}g(x)}{Dg(x)} -\frac{3}{2}(\frac{D^{2}g(x)}{Dg(x)})^{2}.$

Proposition 1. Assume f and g are $C^{3}$ functions, then

$S(f\circ g)(x)= Sf(g(x)) \cdot |Dg(x)|^{2}+ Sg(x).$

$S(f^{n})(x)= \sum_{i=0}^{n-1}(Sf(f^{i}(x)) \cdot |D(f^{i})(x)|^{2}.$

Proposition 2. If $f(x)=x^{\ell}+c$ for some $c\in \mathbb{R}$ and $\ell \geq 2$ , then $Sf(x)<0$ for all $x \neq 0$ .

Proposition 3. Minimum Principle.

Assume $I=[a,b]$ , $f: I \rightarrow \mathbb{R}$ is a $C^{3}$ diffeomorphism with negative schwarzian derivative, then

$|Df(x)| > \min \{|Df(a), |Df(b)|\} \text{ for all } x \in (a,b).$

Theorem 1. Real Koebe Principle.

Let Sf<0. Then for any intervals $j \subseteq t$ and any n for which $f^{n}|t$ is a diffeomorphism one has the following. If $f^{n}(t)$ contains a $\tau-$ scaled neighbourhood of $f^{n}(j)$ , then

$(\frac{\tau}{1+\tau})^{2} \leq \frac{|Df^{n}(x)|}{|Df^{n}(y)|} \leq (\frac{1+\tau}{\tau})^{2} \text{ for all } x, y \in j.$

Moreover, there exists a universal function $K(\tau)>0$ which does not depend on f, n, and t such that

$|l| / |j| \geq K(\tau),$

$|r| /|j| \geq K(\tau).$

Theorem 2. Complex Koebe Principle

Suppose that $D \subseteq \mathbb{C}$ contains a $\tau-$ scaled neighbourhood of the disc $D_{1} \subseteq \mathbb{C}$ . Then for any univalent function $f: D \rightarrow \mathbb{C}$ one has a universal function $K(\tau)>0$ which only depends on $\tau>0$ such that

$1/K(\tau) \leq \frac{|Df(x)|}{|Df(y)|} \leq K(\tau) \text{ for all } x, y \in D_{1}.$

Theorem 3. Schwarz Lemma (Original Form)

Assume $\mathbb{D}=\{ z: |z|<1\}$ is the unit disc on the complex plane $\mathbb{C}$ , $f: \mathbb{D} \rightarrow \mathbb{D}$ is a holomorphic function with $f(0)=0$ . Then $|f(z)|\leq |z|$ for all $z \in \mathbb{D}$ and $|f^{'}(0)| \leq 1.$ Moreover, if $|f(z_{0})|=|z_{0}|$ for some $z_{0}\neq 0$ or $|f^{'}(0)|=1,$ then $f(z)= e^{i\theta} z$ for some $\theta \in \mathbb{R}.$

Corollary 1.

Assume $\mathbb{D}$ is the unit disc on the complex plane $\mathbb{C}$ , and $f: \mathbb{D} \rightarrow \mathbb{D}$ is a holomorphic function, then

$|\frac{f(z)-f(z_{0})}{1-\overline{f(z_{0})}f(z)}| \leq |\frac{z-z_{0}}{1-\overline{z}_{0}z}| \text{ for all } z, z_{0} \in \mathbb{D}.$

$\frac{|f^{'}(z)|}{1-|z|^{2}} \leq \frac{1}{1-|z|^{2}} \text{ for all } z \in \mathbb{D}.$

Corollary 2.

Assume $\mathbb{H}$ is the upper half plane of the complex plane $\mathbb{C}$ , $f: \mathbb{H} \rightarrow \mathbb{H}$ is a holomorphic map. Then

$\frac{|f(z_{1})-f(z_{2})|}{|f(z_{1})-\overline{f(z_{2})}|} \leq \frac{|z_{1}-z_{2}|}{|z_{1}-\overline{z_{2}}|} \text{ for all } z_{1}, z_{2} \in \mathbb{H} .$

$\frac{|f^{'}(z)|}{\Im{f(z)}} \leq \frac{1}{\Im{z}} \text{ for all } z\in \mathbb{H} .$

Corollary 3. Pick Theorem

The hyperbolic metric on $\mathbb{D}$ is $\rho(z)= \frac{2}{1-|z|^{2}}dz$ , assume $d(z_{1}, z_{2})$ denotes the hyperbolic distance between $z_{1}$ and $z_{2}$ on $\mathbb{D}$ . Assume $f: \mathbb{D} \rightarrow \mathbb{D}$ is a holomorphic function, then

$d(f(z_{1}), f(z_{2}))\leq d(z_{1}, z_{2}) \text{ for all } z_{1}, z_{2} \in \mathbb{D}.$

Moreover, if $d(f(z_{1}), f(z_{2}))= d(z_{1}, z_{2})$ for some points $z_{1}, z_{2} \in \mathbb{D}$ , then $f \in Aut(\mathbb{D})$ , where

$Aut(\mathbb{D})=\{e^{i\theta}\frac{z-z_{0}}{1-\overline{z_{0}}z}: \theta \in \mathbb{R}, z_{0} \in \mathbb{D}\}.$

Background in hyperbolic geometry

Define

$\mathbb{C}_{J}=(\mathbb{C} \setminus \mathbb{R}) \cup J$

where $J \subseteq \mathbb{R}$ is an interval. It is easy to show that $\mathbb{C}_{J}$ is conformally equivalent to the upper half plane and define $D_{k}(J)$ as

$D_{k}(J)= \{ z: \text{the hyperbolic distance to J is at most k} \}.$

k is determined by the external angle $\alpha$ at which the discs intersect the real line. Moreover, $k=\ln \tan( \frac{\pi}{2}- \frac{\alpha}{4}) .$ Define

$D_{*}(J)=D(J,\frac{\pi}{2}) .$

Corollary 4. (NS) Schwarz Lemma

(1) Assume $G: \mathbb{C}_{I} \rightarrow \mathbb{C}_{J}$ is a holomorphic map, then $G((D_{*}{I})) \subseteq D_{*}(J).$

(2) Assume $F: \mathbb{C} \rightarrow \mathbb{C}$ is a real polynomial map, its critical points are on the real line. Assume $F: I \rightarrow J$ is a diffeomorphism, then there exists a set $D \subseteq D_{*}(I)$ such that $D\cap \mathbb{R} =I$ and

$F: D \rightarrow D_{*}(J)$ is a conformal map.

Corollary 5.

Assume $f: D \rightarrow \mathbb{C}$ is a univalent map and D contains $\tau-$ scaled neighbourhood of $D_{1},$ and assume f maps the real line to the real line. For each $\alpha \in (\pi/2, \pi)$ there exists $\alpha^{'} \in (\alpha, \pi)$ such that if J is a real interval in $D_{1}$ , then

$f(D(J,\alpha)) \supseteq D(f(J), \alpha^{'}).$

The Hyperbolic Metric On the Real Interval and Cross Ratio

As far as we know, the hyperbolic metric on the unit disc $\mathbb{D}=\{|z|<1\}$ is

$\rho_{D}(z)= \frac{2}{1-|z|^{2}}|dz| \text{ for all } z\in \mathbb{D}.$

Then the restriction to the real line is

$\rho_{I}(x)=\frac{2}{1-x^{2}} dx \text{ for all } x \in I=(-1,1).$

Moreover, from it, we can deduce the hyperbolic metric on the real interval $I=(a,b)$ is

$\rho_{I}(x)=\frac{b-a}{(x-a)(b-x)} dx \text{ for all } x \in I=(a,b).$

If $(c,d) \subseteq (a,b)$ , then the hyperbolic length of the interval $(c,d)$ on the total interval $(a,b)$ is

$\ell_{(a,b)}((c,d))=\ell_{t}(j)=\ln(1+Cr(t,j)),$

where $l=(a,c), j=(c,d), r=(d,b), t=(a,b).$

Theorem 4. Assume $f: T \rightarrow f(T) \subseteq \mathbb{R}$ is a $C^{3}$ diffeomorphism with negative schwarzian derivative. Assume $J \subseteq T$ , then

$\ell_{f(T)}(f(J)) \geq \ell_{T}(J).$

That means f expands the hyperbolic metric on the real interval.

Proof. Since the schwarzian derivative of f is negative, $C(f(T),f(J)) \geq C(T,J).$

Therefore, $\ell_{f(T)}(f(J)) \geq \ell_{T}(J).$ That means f expands the hyperbolic metric on the real interval.

Remark. From Schwarz-Pick Theorem, for a holomorphic map $f: \mathbb{D} \rightarrow \mathbb{D}$ , $f$ contracts the hyperbolic distance in the unit disc $\mathbb{D}$ . Conversely, from above, for a $C^{3}$ diffeomorphism $f$ with negative schwarzian derivative, $f$ expands the hyperbolic distance in the real interval.

Exercise 1. “Mathematical Tools for One Dimensional Dynamics” Exercise 6.5, Chapter 6

Let $f: I \rightarrow f(I) \subseteq \mathbb{R}$ be a $C^{3}$ diffeomorphism without fixed points ( $I$ being a closed interval on the real line). If $Sf(x)<0$ for all $x \in I$ , then there exists a unique $x_{0} \in I$ such that $|f(x_{0})-x_{0}| \leq |f(x)-x|$ for all $x \in I$ .

Proof. If $f$ is a decreasing map, then the right boundary of the real interval I is the $x_{0}$ . Therefore, assume that $f$ is an increasing map on the real interval I.

Since $f(x)$ has no fixed points on the real interval I, then $f(x)>x$ or $f(x)<x$ for all $x \in I$ . Without lost of generality, assume $f(x)>x$ for all $x\in I$ . Since $f(x)-x$ is a continuous function on the closed interval I, there exists $x_{0} \in I$ such that $|f(x_{0})-x_{0}| \leq |f(x)-x|$ for all $x\in I$ .

By contradiction, there exist two distinct points $x_{0}, x_{1}$ $(x_{0}<x_{1})$ such that $|f(x_{0})-x_{0}| \leq |f(x)-x|$ and $|f(x_{1})-x_{1}| \leq |f(x)-x|$ for all $x\in I$ . From here, we know that $|f(x_{0})-x_{0}|= |f(x_{1})-x_{1}|$ .

From Langrange’s mean value theorem, there exists $\xi \in (x_{0}, x_{1})$ such that $(Df)(\xi)=1$ . Since the schwarzian derivative of $f$ is negative, from the minimal principle, we get

$(Df)(\xi) > \min(Df(x_{0}), Df(x_{1})).$

i.e. $Df(x_{0})<1, Df(x_{1})<1$ . However, from the definition of $x_{0}$ and $x_{1}$ , we get

$Df(x_{0}) = \lim_{x\rightarrow x_{0}^{+}} \frac{f(x)-f(x_{0})}{x-x_{0}} \geq 1$

$Df(x_{1}) = \lim_{x\rightarrow x_{1}^{-}} \frac{f(x_{1})-f(x)}{x_{1}-x} \leq 1$

This is a contradiction. Therefore, the existence of $x_{0}$ is unique.

Assume $f: I \rightarrow f(I) \subseteq \mathbb{R}$ is a $C^{3}$ diffeomorphism, define the non-linearity of $f$ as

$f \mapsto Nf=\frac{D^{2}f}{Df} = D \ln Df \text{ whenever } Df \neq 0.$

Proposition 4. $N(f \circ g)= (Nf \circ g) \cdot Dg+ Ng.$

Proposition 5. $Sf=\frac{D^{3}f}{Df} -\frac{3}{2}(\frac{D^{2}f}{Df})^{2}=D(Nf)-\frac{1}{2}(Nf)^{2}.$

Theorem 5. Koebe Non-linearity Principle.

Given $B, \tau>0$ , there exists $K_{\tau,B}>0$ such that, if $f: [-\tau, 1+\tau] \rightarrow \mathbb{R}$ is a $C^{3}$ diffeomorphism into the reals and $Sf(t)\geq -B$ for all $t\in [-\tau,1+\tau],$ then we have