# 统计与大数据专业究竟在学什么？

### 参考资料：

2. http://www.stat.nus.edu.sg：NUS 统计系官网。

# 傅里叶分析与调和分析

### 调和分析

(1) 算子插值方法

(2) Hardy—Littlewood Maximal Operator

(3) Fourier Transformation

(4) Calderon-Zygmund’s Inequality

(5) 函数空间

(6) Ap weight

ps：去 BICMR 学习是 2009 年的事了，一晃眼 11 年过去了。

### 参考文献：

1.  Loukas Grafakos，GTM249，Classical Fourier Analysis
2. Loukas Grafakos，GTM250，Modern Fourier Analysis
3. Elias M.Stein 傅里叶分析导论

1. 第 1，2 两本书是调和分析的经典之作，几乎涵盖了实变方法的所有内容。不过有点厚，差不多 1100 页。
2. 除此之外，也可以阅读 Elias M.Stein 所撰写的《调和分析》，但是这本书不适合做教材，只能够作为翻阅的材料进行阅读。

# 在新加坡的这五年—学术篇

### 基础书籍：

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

(2) Complex Analysis, Elias M. Stein

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

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

(1) Real Analysis and Complex Analysis, Rudin

(2) Real Analysis, Elias M. Stein

### 专业书籍：

(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

(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

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

### 论文

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？

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

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

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

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

### 参考资料：

1. 科研这条路
2. 维基百科：Julia 集合

# 一个数学家的辩白

《一个数学家的辩白》是数学大师 Godfrey Harold Hardy 在 1940 年左右的作品，可以称之为 Hardy 本人的自传和内心独白。Hardy 通过自己多年从事数学科研的经验，对数学，文学，哲学，美学等诸多学科的理解，将其写成了一本小册子。给其余的数学工作者，数学爱好者，以及不了解数学的人一个了解数学家内心的机会。

G.H.Hardy（1877 年 2 月 7 日 – 1947 年 12 月 1 日）是一代数论大师，英国数学家，先后在牛津大学（Oxford）和剑桥大学（Cambridge）担任数学教授，与另一位英国数学家 Littlewood 共同研究数学，其研究领域包括解析数论，三角级数，不等式等诸多方向。对数论领域和分析学领域的贡献巨大，是二十世纪英国分析学派的代表人物之一。

  在这平坦的沙滩上，
海洋与大地间，
我该建起或写些什么，
来阻止夜幕的降临?
告诉我神秘的字符，
去喝退那汹涌的波涛，
告诉我时间的城堡，
去规划那更久的白昼。

 倾江海之水，洗不净帝王身上的膏香御气。

1. 《一个数学家的辩白》；
2. 《黎曼猜想漫谈》。

# 一代数学大师 Rota 的经验与忠告

### Ten Lessons I wish I Had Been Taught

#### 讲座（Lecturing）

（a）每次讲座都应该只有一个重点。（Every lecture should make only one main point.）

Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

（b）不要超时。（Never run overtime.）

Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one micro-century as von Neumann used to say) everybody’s attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

As you enter the lecture hall, try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person’s work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.

Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.

（d）给听众一些值得回忆的东西。（Give them something to take home.）

Most of the time they admit that they have forgotten the subject of the course and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.

#### 板书技巧（Blackboard Technique）

（a）开讲前保持黑板干净（Make sure the blackboard is spotless.）

By starting with a spotless blackboard you will subtly convey the impression that the lecture they are about to hear is equally spotless.

（b）从黑板的左上角开始书写（Start writing on the top left-hand corner.

What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations.

When slides are used instead of the blackboard, the speaker should spend some time explaining each slide, preferably by adding sentences that are inessential, repetitive, or superfluous, so as to allow any member of the audience time to copy our slide. We all fall prey to the illusion that a listener will find the time to read the copy of the slides we hand them after the lecture. This is wishful thinking.

#### 多次公布同样的结果（Publish the Same Result Several Times）

The mathematical community is split into small groups, each one with its own customs, notation, and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation and who will rightly claim it as his own.

#### 说明性的工作反而更有可能被记得（You Are More Likely to Be Remembered by Your Expository Work）

When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry, and for his text on integral equations.

#### 每个数学家只有少数的招数（Every Mathematician Has Only a Few Tricks）

You admire Erdös’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!

There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.

#### 使用费曼的方法（Use the Feynman Method）

You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”

#### 不要吝啬你的赞美（Give Lavish Acknowledgments）

I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else.

#### 写好摘要（Write Informative Introductions）

If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.

#### 为老年做好心理准备（Be Prepared for Old Age）

You must realize that after reaching a certain age you are no longer viewed as a person. You become an institution, and you are treated the way institutions are treated. You are expected to behave like a piece of period furniture, an architectural landmark, or an incunabulum.

### Ten Lessons for the Survival of a Mathematics Department

#### 不要在其他系讲自己系同事的坏话（Never wash your dirty linen in public）

Departments of a university are like sovereign states: there is no such thing as charity towards one another.

Your letter will be viewed as evidence of disunity in the rank and file of mathematicians. Human nature being what it is, such a dean or provost is likely to remember an unsolicited letter at budget time, and not very kindly at that.

#### 不要进行领域评价（Never Compare Fields）

You are not alone in believing that your own field is better and more promising than those of your colleagues. We all believe the same about our own fields. But our beliefs cancel each other out. Better keep your mouth shut rather than make yourself obnoxious. And remember, when talking to outsiders, have nothing but praise for your colleagues in all fields, even for those in combinatorics. All public shows of disunity are ultimately harmful to the well-being of mathematics.

#### 别看不起别人使用的数学（Remember that the grocery bill is a piece of mathematics too）

The grocery bill, a computer program, and class field theory are three instances of mathematics. Your opinion that some instances may be better than others is most effectively verbalized when you are asked to vote on a tenure decision. At other times, a careless statement of relative values is more likely to turn potential friends of mathematics into enemies of our field. Believe me, we are going to need all the friends we can get.

#### 善待擅长教学的老师（Do not look down on good teachers）

Mathematics is the greatest undertaking of mankind. All mathematicians know this. Yet many people do not share this view. Consequently, mathematics is not as self-supporting a profession in our society as the exercise of poetry was in medieval Ireland. Most of our income will have to come from teaching, and the more students we teach, the more of our friends we can appoint to our department. Those few colleagues who are successful at teaching undergraduate courses should earn our thanks as well as our respect. It is counterproductive to turn up our noses at those who bring home the dough.

#### 学会推销自己的数学成果（Write expository papers）

When I was in graduate school, one of my teachers told me, “When you write a research paper, you are afraid that your result might already be known; but when you write an expository paper, you discover that nothing is known.”

It is not enough for you (or anyone) to have a good product to sell; you must package it right and advertise it properly. Otherwise you will go out of business.

When an engineer knocks at your door with a mathematical question, you should not try to get rid of him or her as quickly as possible.

#### 不要把提问者拒之门外（Do not show your questioners to the door）

What the engineer wants is to be treated with respect and consideration, like the human being he is, and most of all to be listened to with rapt attention. If you do this, he will be likely to hit upon a clever new idea as he explains the problem to you, and you will get some of the credit.

Listening to engineers and other scientists is our duty. You may even learn some interesting new mathematics while doing so.

#### 联合阵线（View the mathematical community as a United Front）

Grade school teachers, high school teachers, administrators and lobbyists are as much mathematicians as you or Hilbert. It is not up to us to make invidious distinctions. They contribute to the well-being of mathematics as much as or more than you or other mathematicians. They are right in feeling left out by snobbish research mathematicians who do not know on which side their bread is buttered. It is our best interest, as well as the interest of justice, to treat all who deal with mathematics in whatever way as equals. By being united we will increase the probability of our survival.

#### 把科学从不可靠中拯救出来（Attack Flakiness）

Flakiness is nowadays creeping into the sciences like a virus through a computer, and it may be the present threat to our civilization. Mathematics can save the world from the invasion of the flakes by unmasking them and by contributing some hard thinking. You and I know that mathematics is not and will never be flaky, by definition.

This is the biggest chance we have had in a long while to make a lasting contribution to the well-being of Science. Let us not botch it as we did with the few other chances we have had in the past.

#### 善待所有人（Learn when to withdraw）

Let me confess to you something I have told very few others (after all, this message will not get around much): I have written some of the papers I like the most while hiding in a closet. When the going gets rough, we have recourse to a way of salvation that is not available to ordinary mortals: we have that Mighty Fortress that is our Mathematics. This is what makes us mathematicians into very special people. The danger is envy from the rest of the world.

When you meet someone who does not know how to differentiate and integrate, be kind, gentle, understanding. Remember, there are lots of people like that out there, and if we are not careful, they will do away with us, as has happened many times before in history to other Very Special People.

### 参考资料：

1. Rota, Gian-Carlo. “Ten lessons I wish I had been taught.” Indiscrete thoughts. Birkhäuser, Boston, MA, 1997. 195-203.
2. Rota, Gian-Carlo. “Ten Lessons for the Survival of a Mathematics Department.” Indiscrete Thoughts. Birkhäuser, Boston, MA, 1997. 204-208.

# 素数之美

### 哥德巴赫猜想（Goldbach’s Conjecture）

1. [Theorem] 每一个大于 7 的奇数都可以写成三个素数之和；
2. [Conjecture] 每一个大于 6 的偶数都可以写成两个素数之和。

[Theorem (Vinogradov)] 假设 $N$ 是一个奇数，令 $r(N) = \sum_{p_{1}+p_{2}+p_{3}=N}1$ 表示关于 $N$ 的计数函数，其中 $p_{1}, p_{2}, p_{3}$ 都是素数。则存在一个一致有界的函数 $\Omega(N) \in (c_{1},c_{2})$$c_{2}>c_{1}>0$）对于充分大的奇数 $N$，有以下式子成立

$r(N) = \Omega(N)\cdot \frac{N^{2}}{(\ln(N))^{3}}\cdot\bigg\{1+O\bigg(\frac{\ln\ln(N)}{\ln(N)}\bigg)\bigg\}.$

[Theorem (Chen)] 假设 $N$ 是一个偶数，令 $r(N)=\sum_{p+n=N}1$ 表示关于 $N$ 的计数函数，其中 $p$ 是素数，$n$ 表示最多为两个素数的乘积。则当 $n$ 充分大的时候，有以下式子成立：

$r(N) >> \Omega(N)\cdot \frac{2N}{(\ln(N))^{2}},$

1. 在哥德巴赫猜想的研究过程中，通常数学家把偶数可表示为 $a$ 个素数的乘积与 $b$ 个素数的乘积之和这个问题，简称为 $a + b$ 问题。所以，陈景润证明的 “1+2” 并不是指 1+2 = 3，而指的是对于每一个充分大的偶数，要么是两个素数之和，要么是一个素数加上两个素数之积。其实可以简单的理解为 $p_{1}+p_{2}$ 或者 $p_{1}+p_{2}\cdot p_{3}$，在这里 $p_{1},p_{2},p_{3}$ 都是素数。从以上公式可以看出，$\lim_{N\rightarrow \infty} r(N) = +\infty.$
2. 1920 年，挪威数学家 V.Brun 证明了 “9+9″，开启了数学家研究哥德巴赫猜想之路；1966 年，中国数学家陈景润证明了 “1+2″，把素数的筛法推向了顶峰。

### 孪生素数猜想（Twin Primes Conjecture）

1. 1940 年，Paul Erdos 证明 $\exists c>0$ 使得 $\liminf_{n\rightarrow\infty} \frac{p_{n+1}-p_{n}}{\ln(p_{n})}
2. 2005 年，Daniel Goldston，Janos Pintz 和 Cem Yildirim 证明 $\liminf_{n\rightarrow\infty}\frac{p_{n+1}-p_{n}}{\ln(p_{n})}=0.$
3. 2007 年，上述结果被改进为 $\liminf_{n\rightarrow\infty}\frac{p_{n+1}-p_{n}}{\sqrt{\ln(p_{n})}\cdot (\ln\ln(p_{n}))^{2}}=0.$
4. 2013 年，张益唐证明了 $\liminf_{n\rightarrow\infty}(p_{n+1}-p_{n}) < 7 * 10^{7}$，随后这个结果被改进到 246。

1. 1931 年，Erik Westzynthius 证明 $\limsup_{n\rightarrow\infty}\frac{p_{n+1}-p_{n}}{\ln(p_{n})} =\infty.$
2. 2014 年，Kevin Ford, Ben Green, Sergei Konyagin, Terence Tao 和 James Maynard 证明 $p_{n+1}-p_{n}>c\cdot \frac{\ln(n)\cdot \ln\ln(n) \cdot \ln\ln\ln\ln(n)}{\ln\ln\ln(n)}$ 对于某个 $c>0$ 和无穷个 $n$ 成立。

### 素数定理

[素数定理] 假设 $\pi(x)$ 表示不大于 $x$ 的所有素数的个数，那么 $\lim_{x\rightarrow \infty}\pi(x)/(x/\ln(x)) = 1.$

[孪生素数个数的上界] 假设 $\pi_{2}(x)$ 表示不大于 $x$ 的所有孪生素数个数，那么存在常数 $C>0$ 使得 $\pi_{2}(x)\leq C\cdot x/(\ln(x))^{2}.$

## 素数的性质

[Theorem (Euclid)] 素数有无穷多个。

$S(x) = \sum_{1\leq n\leq x} \frac{1}{n}.$

$\sum_{n=1}^{\infty}\frac{1}{n} = \prod_{p\text{ prime}}\bigg(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\bigg)= \prod_{p\text{ prime}} \frac{1}{1-\frac{1}{p}},$

$\ln\bigg(\sum_{n=1}^{\infty}\frac{1}{n} \bigg)= \sum_{p\text{ prime}}\frac{1}{p} + O\bigg(\sum_{p\text{ prime}}\frac{1}{p^{2}}\bigg).$

1. $\sum_{p\text{ prime}, p\leq x} \frac{1}{p} = \ln\ln(x) + O(1);$
2. $\prod_{p\text{ prime}, p\leq x}\bigg(1-\frac{1}{p}\bigg)^{-1} = c\cdot\ln(x)+o(1),$ 这里，$c>0$ 是一个常数。

1. [Theorem] 所有正整数的倒数和是发散的；
2. [Theorem] 所有素数的倒数和是发散的。

[Theorem] 所有孪生素数的倒数和是收敛的。

$\pi_{2}(x) \leq C\cdot \frac{x}{(\ln(x))^{2}}$

$\frac{1}{p_{n}'} \leq C\frac{1}{n\cdot (\ln(n))^{2}}$

[Theorem] 对于充分大的 $x$ 而言，在 $[1,x]$ 内，素数之间的最小间隔 $\min_{p_{n}\leq x} (p_{n}-p_{n-1})\leq (1+o(1))\ln(x);$ 同时，素数之间的最大间隔 $\max_{p_{n}\leq x}(p_{n}-p_{n-1})\geq (1+o(1))\ln(x).$

### Eratosthenes 筛法（Eratosthenes Sieve Method)

Eratosthenes 筛法是数学家 Eratosthenes 提出的一种筛选素数的方法，其思路比较简单：想要筛选出 $[2,n]$ 中的所有素数，则首先把 $[2,n]$ 中的所有正整数按照从小到大的顺序 $2, \cdots, n$ 来排列，然后按照如下步骤执行：

1. 读取数列中当前最小的数 2，然后把 2 的倍数全部删除；
2. 读取数列中当前最小的数 3，然后把 3 的倍数全部删除；
3. 读取数列中当前最小的数 5，然后把 5 的倍数全部删除；（4 已经被第一步去掉了）
4. 读取数列中当前最小的数 7，然后把 7 的倍数全部删除；（6 已经被第一步去掉了）
5. 循环以上步骤直到 $[2,n]$ 中所有的数被读取或者被删除。

### Brun 筛法（Brun Sieve Method）

Question. 研究素数究竟有什么用？

### 参考文献：

1.  Small and Large Gaps Between Primes, Terence Tao, Latinos in the Mathematical Sciences Conference, 2015.
2.  Bounded Gaps Beween Primes, Yitang Zhang, 2013.
3.  Additive Number Theory, Melvyn B.Nathanson, GTM 164.
4.  http://mathworld.wolfram.com/TwinPrimes.html.

# 拉格朗日四平方和定理

### 拉格朗日四平方和定理

Every positive integer is the sum of four squares.

• $1=1^{2}+0^{2}+0^{2}+0^{2}$
• $2 = 1^{2}+1^{2}+0^{2}+0^{2}$
• $7 = 2^{2}+1^{2}+1^{2}+1^{2}$

$(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\cdot(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}) = z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}$，其中

$\begin{cases} z_{1}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4} \\ z_{2}=x_{1}y_{2}-x_{2}y_{1}-x_{3}y_{4}+x_{4}y_{3} \\ z_{3}=x_{1}y_{3}-x_{3}y_{1}+x_{2}y_{4}-x_{4}y_{2} \\ z_{4}=x_{1}y_{4}-x_{4}y_{1}-x_{2}y_{3}+x_{3}y_{2}\end{cases}$

Claim. $m=1$

proof of the claim. 反证法，假设 $1 成立。令 $y_{i}=x_{i}(\mod m)$ 对于 $i\in\{1,2,3,4\}$ 成立，并且 $-m/2。因此，$y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}\equiv(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\equiv mp \equiv 0(\mod m)$。令 $mr = y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}$。因此，$mr\leq 4(m/2)^{2}=m^{2}$

### 参考文献

1. GTM 164, Additive Number Theory, Melvyn B.Nathanson, 1996.

# 传染病的数学模型

• 易感者（susceptible）：用符号 $S(t)$ 来表示；
• 感染者（infective）：用符号 $I(t)$ 来表示；
• 康复者（Recoverd）：用符号 $R(t)$ 来表示；

• $r$ 表示在单位时间内感染者接触到的易感者人数；
• 传染率：$\beta$ 表示感染者接触到易感者之后，易感者得病的概率；
• 康复率：$\gamma$ 表示感染者康复的概率，有可能变成易感者（可再感染），也有可能变成康复者（不再感染）。

Claim. 假设 $x=x(t)$ 是关于 $t$ 的一个方程，且满足 $\frac{dx}{dt} + a_{1}x + a_{2}x^{2}=0$$x(0)=x_{0}$，那么它的解是：$x(t) = \frac{e^{-a_{1}t}}{\frac{1}{x_{0}}-\frac{a_{2}}{a_{1}}(e^{-a_{1}t}-1)}$.

Proof. 证明如下：

### SI 模型（Susceptible-Infective Model）

$\begin{cases}\frac{dS}{dt} = -\frac{r\beta I}{N} S \\ \frac{dI}{dt}=\frac{r\beta I}{N}S \end{cases}$

$I(t) = \frac{NI_{0}}{I_{0}+(N-I_{0})e^{-r\beta t}}$.

### SIS 模型（Susceptible-Infectious-Susceptible Model）

$\begin{cases} \frac{dS}{dt} = -r \beta S\frac{I}{N} + \gamma I \\ \frac{dI}{dt}=r\beta S \frac{I}{N} - \gamma I \end{cases}$，其初始条件就是 $S(0)=S_{0}$$I(0)=I_{0}$.

$I(t) = \frac{N(r\beta-\gamma)}{r\beta}/\bigg(\bigg(\frac{N(r\beta-\gamma)}{I_{0}r\beta}-1\bigg)e^{-(r\beta-\gamma)t}+1\bigg)$.

### 参考文献

1. Introduction to SEIR Models, Nakul Chitnis, Workshop on Mathematical Models of Climate Variability, Environmental Change and Infectious Diseases, Trieste, Italy, 2017

# SymPy 的简单介绍

SymPy 是一个符号计算的 Python 库，完全由 Python 写成，为许多数值分析，符号计算提供了重要的工具。SymPy 的第一个版本于 2007 年开源，并且经历了十几个版本的迭代，在 2019 年已经基于修正的 BSD 许可证开源了 1.4 版本。SymPy 的开源地址和官方网站分别是：

1. GitHub 链接：https://github.com/sympy/sympy
2. SymPy 官方网站：https://www.sympy.org/en/index.html

SymPy 的 1.4 版本文档中，可以看出，SymPy 可以支持很多初等数学，高等数学，甚至研究生数学的符号计算。在初等数学和高等数学中，SymPy 可以支持的内容包括但不限于：

1. 基础计算（Basic Operations）；
2. 公式简化（Simplification）；
3. 微积分（Calculus）；
4. 解方程（Solver）；
5. 矩阵（Matrices）；
6. 几何（geometry）；
7. 级数（Series）；

1. 范畴论（Category Theory）；
2. 微分几何（Differential Geometry）；
3. 常微分方程（ODE）；
4. 偏微分方程（PDE）；
5. 傅立叶变换（Fourier Transform）；
6. 集合论（Set Theory）；
7. 逻辑计算（Logic Theory）。

# SymPy 的工具库介绍

## SymPy 的基础计算

$e^{i\pi}+1 = 0$

sympy.exp(1), sympy.I, sympy.pi, sympy.oo

>>> sympy.exp(sympy.I * sympy.pi) + 1
0

>>> sympy.E.evalf(10)
2.718281828
>>> sympy.E.evalf()
2.71828182845905
>>> sympy.pi.evalf(10)
3.141592654
>>> sympy.pi.evalf()
3.14159265358979

>>> expr = sympy.sqrt(8)
>>> expr.evalf()
2.82842712474619

>>> x, y= sympy.symbols("x y")
>>> x + y
x + y
>>> x - y
x - y
>>> x * y
x*y
>>> x / y
x/y

>>> x1, y1, x2, y2 = sympy.symbols("x1 y1 x2 y2")
>>> z1 = x1 + y1 * sympy.I
x1 + I*y1
>>>  z2 = x2 + y2 * sympy.I
x2 + I*y2
>>> z1 + z2
x1 + x2 + I*y1 + I*y2
>>> z1 - z2
x1 - x2 + I*y1 - I*y2
>>> z1 * z2
(x1 + I*y1)*(x2 + I*y2)
>>> z1 / z2
(x1 + I*y1)/(x2 + I*y2)

>>> sympy.expand((x+1)**2)
x**2 + 2*x + 1
>>> sympy.expand((x+1)**5)
x**5 + 5*x**4 + 10*x**3 + 10*x**2 + 5*x + 1
>>> sympy.factor(x**3+1)
(x + 1)*(x**2 - x + 1)
>>> sympy.factor(x**2+3*x+2)
(x + 1)*(x + 2)
>>> sympy.simplify(x**2 + x + 1 - x)
x**2 + 1
>>> sympy.simplify(sympy.sin(x)**2 + sympy.cos(x)**2)
1


>>> expr = x*y + x - 3 + 2*x**2 - x**2 + x**3 + y**2 + x**2*y**2
>>> sympy.collect(expr,x)
x**3 + x**2*(y**2 + 1) + x*(y + 1) + y**2 - 3
>>> sympy.collect(expr,y)
x**3 + x**2 + x*y + x + y**2*(x**2 + 1) - 3
>>> expr.coeff(x, 2)
y**2 + 1
>>> expr.coeff(y, 1)
x

$expr = \frac{x^{2}+3x+2}{x^{2}+x}$

>>> expr = (x**2 + 3*x + 2)/(x**2 + x)
>>> sympy.cancel(expr)
(x + 2)/x
>>> sympy.together(expr)
(x**2 + 3*x + 2)/(x*(x + 1))

expr = (x**2 + 3*x + 2)/(x**2 + x)
>>> sympy.factor(expr)
(x + 2)/x
>>> expr = (x**3 + 3*x**2 + 2*x)/(x**5+x)
>>> sympy.factor(expr)
(x + 1)*(x + 2)/(x**4 + 1)
>>> expr = x**2 + (2*x+1)/(x**3+1)
>>> sympy.factor(expr)
(x**5 + x**2 + 2*x + 1)/((x + 1)*(x**2 - x + 1))

>>> expr = (x**4 + 3*x**2 + 2*x)/(x**2+x)
>>> sympy.apart(expr)
x**2 - x + 4 - 2/(x + 1)
>>> expr = (x**5 + 1)/(x**3+1)
>>> sympy.apart(expr)
x**2 - (x - 1)/(x**2 - x + 1)

>>> expr = sympy.sin(x)**2 + sympy.cos(x)**2
>>> sympy.trigsimp(expr)
1
>>> sympy.expand_trig(sympy.sin(x+y))
sin(x)*cos(y) + sin(y)*cos(x)
>>> sympy.expand_trig(sympy.cos(x+y))
-sin(x)*sin(y) + cos(x)*cos(y)
>>> sympy.trigsimp(sympy.sin(x)*sympy.cos(y) + sympy.sin(y)*sympy.cos(x))
sin(x + y)
>>> sympy.trigsimp(-sympy.sin(x)*sympy.sin(y) + sympy.cos(x)*sympy.cos(y))
cos(x + y)

>>> sympy.powsimp(x**z*y**z*x**z)
x**(2*z)*y**z
>>> sympy.simplify(x**z*y**z*x**z)
x**(2*z)*y**z
>>> sympy.expand_power_exp(x**(y + z))
x**y*x**z
>>> sympy.expand_power_base(x**(y + z))
x**(y + z)

$\ln(xy) = \ln(x) + \ln(y)$

$\ln(x/y) = \ln(x) - \ln(y)$

>>> sympy.expand_log(sympy.log(x*y), force=True)
log(x) + log(y)
>>> sympy.expand_log(sympy.log(x/y), force=True)
log(x) - log(y)

## SymPy 的微积分工具

>>> import sympy
>>> x = sympy.Symbol("x")
>>> f = 1 / x
1/x
>>> y = sympy.Symbol("y")
>>> f = f.subs(x,y)
1/y
>>> f = f.subs(y,1)
1

>>> f = 1/x
>>> sympy.limit(f,x,0)
oo
>>> sympy.limit(f,x,2)
1/2
>>> sympy.limit(f,x,sympy.oo)
0
>>> g = x * sympy.log(x)
>>> sympy.limit(g,x,0)
0

>>> f = 1/x
>>> sympy.diff(f,x)
-1/x**2
>>> sympy.diff(f,x,2)
2/x**3
>>> sympy.diff(f,x,3)
-6/x**4
>>> sympy.diff(f,x,4)
24/x**5

sympy.series.series.series(expr, x=None, x0=0, n=6, dir='+') >>> g = sympy.cos(x) >>> sympy.series(g, x) 1 - x**2/2 + x**4/24 + O(x**6) >>> sympy.series(g, x, x0=1, n=10) cos(1) - (x - 1)*sin(1) - (x - 1)**2*cos(1)/2 + (x - 1)**3*sin(1)/6 + (x - 1)**4*cos(1)/24 - (x - 1)**5*sin(1)/120 - (x - 1)**6*cos(1)/720 + (x - 1)**7*sin(1)/5040 + (x - 1)**8*cos(1)/40320 - (x - 1)**9*sin(1)/362880 + O((x - 1)**10, (x, 1))

$\int\frac{1}{x}dx = \ln(x)+C$

$\int_{1}^{2}\frac{1}{x}dx = \ln(2)$

>>> f = 1/x
>>> sympy.integrate(f,x)
log(x)
>>> sympy.integrate(f, (x,1,2))
log(2)

$\int_{-\infty}^{0}e^{-x^{2}}dx=\frac{\sqrt{\pi}}{2}$

$\int_{0}^{+\infty}e^{-x}dx = 1$

$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-x^{2}-y^{2}}dxdy = \pi$

>>> g = sympy.exp(-x**2)
>>> sympy.integrate(g, (x,-sympy.oo,0))
sqrt(pi)/2
>>> g = sympy.exp(-x)
>>> sympy.integrate(g, (x, 0, sympy.oo))
1
>>> h = sympy.exp(-x**2 - y**2)
>>> sympy.integrate(h, (x,-sympy.oo, sympy.oo), (y, -sympy.oo, sympy.oo))
pi

## SymPy 的方程工具

$\{x\in\mathbb{R}: x^{3}-1=0\}$

$\{x\in\mathbb{C}:x^{3}-1=0\}$

$\{x\in\mathbb{R}:e^{x}-x=0\}$

$\{x\in\mathbb{R}:e^{x}-1=0\}$

$\{x\in\mathbb{C}:e^{x}-1=0\}$

>>> sympy.solveset(sympy.Eq(x**3,1), x, domain=sympy.S.Reals)
{1}
>>> sympy.solveset(sympy.Eq(x**3,1), x, domain=sympy.S.Complexes)
{1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2}
>>> sympy.solveset(sympy.Eq(x**3 - 1,0), x, domain=sympy.S.Reals)
{1}
>>> sympy.solveset(sympy.Eq(x**3 - 1,0), x, domain=sympy.S.Complexes)
{1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2}
>>> sympy.solveset(sympy.exp(x),x)
EmptySet()
>>> sympy.solveset(sympy.exp(x)-1,x,domain=sympy.S.Reals)
{0}
>>> sympy.solveset(sympy.exp(x)-1,x,domain=sympy.S.Complexes)
ImageSet(Lambda(_n, 2*_n*I*pi), Integers)

$\begin{cases}x+y-10=0 \\ x-y-2=0\end{cases}$

>>> sympy.solve([x+y-10, x-y-2], [x,y])
{x: 6, y: 4}

$\begin{cases} \sin(x-y)=0 \\ \cos(x+y)=0 \end{cases}$

>>> sympy.solve([sympy.sin(x-y), sympy.cos(x+y)], [x,y])
[(-pi/4, 3*pi/4), (pi/4, pi/4), (3*pi/4, 3*pi/4), (5*pi/4, pi/4)]

## SymPy 的矩阵工具

>>> sympy.eye(3)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> sympy.eye(3,2)
Matrix([
[1, 0],
[0, 1],
[0, 0]])
>>> sympy.eye(2,3)
Matrix([
[1, 0, 0],
[0, 1, 0]])

>>> sympy.ones(2,3)
Matrix([
[1, 1, 1],
[1, 1, 1]])
>>> sympy.zeros(3,2)
Matrix([
[0, 0],
[0, 0],
[0, 0]])

>>> sympy.diag(1,1,2)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 2]])

$A = \left(\begin{array}{cc} 1 & 1 \\ 0 & 2 \end{array}\right)$

$B = \left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right)$

>>> A = sympy.Matrix([[1,1],[0,2]])
>>> B = sympy.Matrix([[1,0],[1,1]])
>>> A
Matrix([
[1, 1],
[0, 2]])
>>> B
Matrix([
[1, 0],
[1, 1]])
>>> A + B
Matrix([
[2, 1],
[1, 3]])
>>> A - B
Matrix([
[ 0, 1],
[-1, 1]])
>>> A * B
Matrix([
[2, 1],
[2, 2]])
>>> A * B**-1
Matrix([
[ 0, 1],
[-2, 2]])
>>> B**-1
Matrix([
[ 1, 0],
[-1, 1]])
>>> A.T
Matrix([
[1, 0],
[1, 2]])
>>> A.det()
2

>>> A
Matrix([
[1, 1],
[0, 2]])
>>> A.row_insert(1, sympy.Matrix([[10,10]]))
Matrix([
[ 1, 1],
[10, 10],
[ 0, 2]])
>>> A.col_insert(0, sympy.Matrix([3,3]))
Matrix([
[3, 1, 1],
[3, 0, 2]])
>>> A.row_del(0)
>>> A
Matrix([[0, 2]])
>>> A.col_del(1)
>>> A
Matrix([[0]])

>>> A
Matrix([
[1, 1],
[0, 2]])
>>> A.eigenvals()
{2: 1, 1: 1}
>>> A.eigenvects()
[(1, 1, [Matrix([
[1],
[0]])]), (2, 1, [Matrix([
[1],
[1]])])]
>>> A.diagonalize()
(Matrix([
[1, 1],
[0, 1]]), Matrix([
[1, 0],
[0, 2]]))

>>> A = sympy.Matrix([[1,1],[0,2]])
>>> A
Matrix([
[1, 1],
[0, 2]])
>>> b = sympy.Matrix([3,5])
>>> b
Matrix([
[3],
[5]])
>>> A**-1*b
Matrix([
[1/2],
[5/2]])
>>> sympy.linsolve((A,b))
{(1/2, 5/2)}
>>> sympy.linsolve((A,b),[x,y])
{(1/2, 5/2)}

## SymPy 的集合论工具

I = sympy.Interval(0,1)
J = sympy.Interval.open(0,1)
K = sympy.Interval(0.5,2)

>>> I.start
0
>>> I.end
1

>>> I.measure
1

>>> I.closure
Interval(0, 1)

>>> I.interior
Interval.open(0, 1)

>>> I.left_open
False
>>> I.right_open
False

I = sympy.Interval(0,1)
K = sympy.Interval(0.5,2)
>>> I.intersect(K)
Interval(0.500000000000000, 1)
>>> I.union(K)
Interval(0, 2)
>>> I-K
Interval.Ropen(0, 0.500000000000000)
>>> K-I
Interval.Lopen(1, 2)
>>> I.symmetric_difference(K)
Union(Interval.Ropen(0, 0.500000000000000), Interval.Lopen(1, 2))

>>> sympy.S.Reals
Reals
>>> sympy.S.Reals-I
Union(Interval.open(-oo, 0), Interval.open(1, oo))
>>> I.complement(sympy.S.Reals)
Union(Interval.open(-oo, 0), Interval.open(1, oo))
>>> sympy.S.Reals.complement(I)
EmptySet()
>>> I.complement(K)
Interval.Lopen(1, 2)
>>> I.complement(sympy.S.Reals)
Union(Interval.open(-oo, 0), Interval.open(1, oo))

## SymPy 的逻辑工具

A, B, C = sympy.symbols("A B C")
>>> sympy.simplify_logic(A | (A & B))
A
>>> sympy.simplify_logic((A>>B) & (B>>A))
(A & B) | (~A & ~B)
>>> A>>B
Implies(A, B)

## SymPy 的级数工具

SymPy 的级数工具有一部分放在具体数学（Concrete Mathematics）章节了。有的时候，我们希望计算某个级数是发散的，还是收敛的，就可以使用 is_convergence 函数。考虑最常见的级数：

$\sum_{n=1}^{\infty}\frac{1}{n} = +\infty$

$\sum_{n=1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}$

>>> n = sympy.Symbol("n", integer=True)
>>> sympy.Sum(1/n, (n,1,sympy.oo)).is_convergent()
False
>>> sympy.Sum(1/n**2, (n,1,sympy.oo)).is_convergent()
True

>>> sympy.Sum(1/n**2, (n,1,sympy.oo)).evalf()
1.64493406684823
>>> sympy.Sum(1/n**2, (n,1,sympy.oo)).doit()
pi**2/6
>>> sympy.Sum(1/n**3, (n,1,sympy.oo)).evalf()
1.20205690315959
>>> sympy.Sum(1/n**3, (n,1,sympy.oo)).doit()
zeta(3)

$\prod_{n=1}^{+\infty}\frac{n}{n+1}$

$\prod_{n=1}^{+\infty}\cos\left(\frac{\pi}{n}\right)$

>>> sympy.Product(n/(n+1), (n,1,sympy.oo)).is_convergent()
False
>>> sympy.Product(sympy.cos(sympy.pi/n), (n, 1, sympy.oo)).is_convergent()
True

## SymPy 的 ODE 工具

$df/dx + f(x) = 0$,

$d^{2}f/dx^{2} + f(x) = 0$

$d^{3}f/dx^{3} + f(x) = 0$

>>> f = sympy.Function('f')
>>> sympy.dsolve(sympy.Derivative(f(x),x) + f(x), f(x))
Eq(f(x), C1*exp(-x))
>>> sympy.dsolve(sympy.Derivative(f(x),x,2) + f(x), f(x))
Eq(f(x), C1*sin(x) + C2*cos(x))
>>> sympy.dsolve(sympy.Derivative(f(x),x,3) + f(x), f(x))
Eq(f(x), C3*exp(-x) + (C1*sin(sqrt(3)*x/2) + C2*cos(sqrt(3)*x/2))*sqrt(exp(x)))

>>> sympy.classify_ode(sympy.Derivative(f(x),x) + f(x), f(x))
('separable', '1st_exact', '1st_linear', 'almost_linear', '1st_power_series', 'lie_group', 'nth_linear_constant_coeff_homogeneous', 'separable_Integral', '1st_exact_Integral', '1st_linear_Integral', 'almost_linear_Integral')
>>> sympy.classify_ode(sympy.Derivative(f(x),x,2) + f(x), f(x))
('nth_linear_constant_coeff_homogeneous', '2nd_power_series_ordinary')
>>> sympy.classify_ode(sympy.Derivative(f(x),x,3) + f(x), f(x))
('nth_linear_constant_coeff_homogeneous',)

## SymPy 的 PDE 工具

$\partial f/\partial x + \partial f/\partial y =0$

$\partial f/\partial x + \partial f/\partial y + f = 0$

$\partial f/\partial x + \partial f/\partial y + f + 10 = 0$

>>> f = sympy.Function("f")(x,y)
>>> sympy.pdsolve(sympy.Derivative(f,x)+sympy.Derivative(f,y),f)
Eq(f(x, y), F(x - y))
>>> sympy.pdsolve(f.diff(x)+f.diff(y)+f,f)
Eq(f(x, y), F(x - y)*exp(-x/2 - y/2))
>>> sympy.pdsolve(f.diff(x)+f.diff(y)+f+10,f)
Eq(f(x, y), F(x - y)*exp(-x/2 - y/2) - 10)

>>> sympy.classify_pde(f.diff(x)+f.diff(y)+f)
('1st_linear_constant_coeff_homogeneous',)
>>> sympy.classify_pde(f.diff(x)+f.diff(y)+f+10,f)
('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral')
>>> sympy.classify_pde(f.diff(x)+f.diff(y)+f+10,f)
('1st_linear_constant_coeff', '1st_linear_constant_coeff_Integral')

## SymPy 的数论工具

>>> sympy.sieve._reset()
>>> sympy.sieve.extend_to_no(100)
>>> sympy.sieve._list
array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631])

>>> [i for i in sympy.sieve.primerange(10,100)]
[11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

search() 函数是为了计算某个数附近是第几个素数：

>>> sympy.sieve.search(10)
(4, 5)
>>> sympy.sieve.search(11)
(5, 5)

>>> sympy.ntheory.generate.prime(10)
29
>>> sympy.ntheory.generate.nextprime(10)
11
>>> sympy.ntheory.generate.nextprime(11)
13
>>> sympy.ntheory.generate.isprime(11)
True
>>> sympy.ntheory.generate.isprime(12)
False

## SymPy 的范畴论工具

SymPy 还支持范畴论（Category Theory）的一些计算方法，在这里简要地列举一下。

>>> A = sympy.categories.Object("A")
>>> B = sympy.categories.Object("B")
>>> f = sympy.categories.NamedMorphism(A,B,"f")
>>> f.domain
Object("A")
>>> f.codomain
Object("B")

## 参考文献：

1. Meurer A, Smith C P, Paprocki M, et al. SymPy: symbolic computing in Python[J]. PeerJ Computer Science, 2017, 3: e103.
2. GitHub：https://github.com/sympy/sympy
3. SymPy：https://www.sympy.org/en/index.html
4. Sympy 维基百科：https://en.wikipedia.org/wiki/SymPy
5. GreatX’s Blog：数值 Python：符号计算：https://vlight.me/2018/04/01/Numerical-Python-Symbolic-Computing/
6. SymPy 符号计算-让Python帮我们推公式：https://zhuanlan.zhihu.com/p/83822118
7. Python 科学计算利器—SymPy库：https://www.jianshu.com/p/339c91ae9f41

# 本科学数学专业是一个很好的选择吗？

### 数学专业的劣势

• 理论知识太多；
• 实用技能偏少；
• 转行需要时间。

• 第一年：数学分析，高等代数，解析几何，C++等；
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### 数学专业的优势

• 底层通用技能；
• 技能不易淘汰；
• 逻辑思维能力；
• 转行就业面广。

• 第一年：数学分析，高等代数，解析几何，C++等；
• 第二年：常微分方程，离散数学，复分析，概率论，数值计算，抽象代数等；
• 第三年：实分析，泛函分析，偏微分方程，拓扑学，微分几何，偏微分方程数值解，随机过程，数理统计等。

• 科研工作者：数学界，金融界，经济界，计算机方向等；
• 计算机行业；
• 金融行业；
• 教育培训行业；
• 其他行业。

# Riemann Zeta 函数（二）

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.$

1. $s = 1$ 时，$\zeta(1) = \infty;$
2. $s>1$ 时，$\zeta(s)<\infty.$

1. 如何把 Riemann Zeta 函数从 $[1,\infty)\subseteq \mathbb{R}$ 上延拓到 $\{s\in \mathbb{C}: \Re(s)>0\}$ 上；
2. Riemann Zeta 函数在 $\{s\in\mathbb{C}: \Re(s)\geq 1\}$ 上没有零点。

### Riemann Zeta 函数定义域的延拓

$\zeta(s) = \frac{s}{s-1} - s\int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx$.

$\frac{s}{s-1}-s\int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx$

$= \frac{s}{s-1} - s\sum_{n=1}^{\infty}\int_{n}^{n+1}\frac{\{x\}}{x^{s+1}}dx$

$= \frac{s}{s-1} - s\sum_{n=1}^{\infty}\int_{n}^{n+1}\frac{x-n}{x^{s+1}}dx$

$= \frac{s}{s-1} - s\sum_{n=1}^{\infty}\bigg(\int_{n}^{n+1}\frac{1}{x^{s}}dx - \int_{n}^{n+1}\frac{n}{x^{s+1}}dx\bigg)$

$= \frac{s}{s-1} - s\int_{1}^{\infty}\frac{1}{x^{s}}dx + \sum_{n=1}^{\infty}n\cdot\int_{n}^{n+1}\frac{s}{x^{s+1}}dx$

$= \sum_{n=1}^{\infty}n\cdot\bigg(\frac{1}{n^{s}}-\frac{1}{(n+1)^{s}}\bigg)$

$= \sum_{n=1}^{\infty}\bigg(\frac{1}{n^{s-1}}-\frac{1}{(n+1)^{s-1}} + \frac{1}{(n+1)^{s}}\bigg)$

$= \sum_{n=1}^{\infty}\frac{1}{n^{s}}.$

$\frac{s}{s-1} - s \int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx$

1. Riemann Zeta 函数可以延拓到 $\{s\in\mathbb{C}:\Re(s)>0\}$ 上；
2. Riemann Zeta 函数在 $\{s\in\mathbb{C}:\Re(s)>0, s\neq 1\}$ 上是解析的；$s=1$ 是 Riemann Zeta 函数的极点。

### Riemann Zeta 函数的非零区域

#### $\Re(s)>1$ 区域

$\zeta(s) =\sum_{n=1}^{\infty}\frac{1}{n^{s}}$

$= \prod_{p}\bigg(1+\frac{1}{p^{s}}+\frac{1}{p^{2s}}+\cdots\bigg)$

$= \prod_{n=1}^{\infty}\bigg(1-\frac{1}{p_{n}^{s}}\bigg)^{-1},$

$\bigg|1-\frac{1}{p_{n}^{s}}\bigg|^{-1}\geq 1-\frac{1}{p_{n}^{\sigma}-1} .$

$\bigg|1-\frac{1}{p_{n}^{s}}\bigg|^{-1} = \bigg(1+\frac{1}{p_{n}^{s}}+\frac{1}{p_{n}^{2s}}+\cdots\bigg)$

$\geq 1-\frac{1}{|p_{n}^{s}|}- \frac{1}{|p_{n}^{2s}|} -\cdots$

$= 1- \frac{1}{p_{n}^{\sigma}} - \frac{1}{p_{n}^{2\sigma}} -\cdots$

$= 1- \frac{1}{p_{n}^{\sigma}-1}.$

$|\zeta(s)| \geq \prod_{n=1}^{\infty}\bigg|1-\frac{1}{p_{n}^{s}}\bigg|^{-1} \geq\prod_{n=1}^{\infty}\bigg(1-\frac{1}{p_{n}^{\sigma}-1}\bigg).$

$\lim_{n\rightarrow \infty} \bigg(1- \frac{1}{p_{n}^{\sigma}-1}\bigg) = 1 ,$

$1-\frac{1}{p_{n+1}^{\sigma}-1} \geq 1- \frac{1}{p_{n}^{\sigma}-1} ,$

$\sum_{n=1}^{\infty}\frac{1}{p_{n}^{\sigma}}\leq \sum_{n=1}^{\infty}\frac{1}{n^{\sigma}}<\infty$ when $\sigma>1.$

#### $\Re(s) =1$ 直线

Claim 1. 下面我们将会证明恒等式：对于 $\sigma >1, \text{ } t\in\mathbb{R},$

$\Re(\ln\zeta(\sigma + it)) = \sum_{n=2}^{\infty}\frac{\Lambda(n)}{n^{\sigma}\ln(n)}\cos(t\ln(n)) ,$

$\zeta(s) = \prod_{p}\bigg(1-\frac{1}{p^{s}}\bigg)^{-1}.$

$s = \sigma + it,$ 可以得到

$\ln\zeta(s) = -\sum_{p}\ln\bigg(1-\frac{1}{p^{s}}\bigg)$

$= \sum_{p}\sum_{\alpha=1}^{\infty}\frac{1}{\alpha p^{\alpha s}}$

$= \sum_{p}\sum_{\alpha=1}^{\infty}\frac{1}{\alpha p^{\alpha\sigma}}\cdot p^{-i\alpha t}$

$= \sum_{p}\sum_{\alpha = 1}^{\infty}\frac{1}{\alpha p^{\alpha\sigma}}\cdot e^{-i\alpha t \ln p}$

$\Re(\ln\zeta(s)) = \sum_{p}\sum_{\alpha =1}^{\infty}\frac{1}{\alpha p^{\alpha\sigma}}\cos(\alpha t \ln p)$

$RHS = \sum_{n=2}^{\infty}\frac{\Lambda(n)}{n^{\sigma}\ln(n)}\cos(t\ln(n))$

$= \sum_{p}\sum_{\alpha = 1}^{\infty} \frac{\ln(p)}{p^{\alpha\sigma}\ln(p^{\alpha})}\cos(t\ln(p^{\alpha}))$

$= \sum_{p}\sum_{\alpha = 1}^{\infty}\frac{1}{\alpha p^{\alpha\sigma}}\cos(\alpha t\ln p).$

Claim 2.

$\Re(3\ln\zeta(\sigma) + 4\ln\zeta(\sigma+it) + \ln\zeta(\sigma+2it))\geq 0,$

$|\zeta(\sigma)^{3}\zeta(\sigma+it)^{4}\zeta(\sigma+2it)|\geq 1.$

$3+4\cos(\theta)+\cos(2\theta) = 3 + 4\cos(\theta)+2\cos^{2}(\theta)-1$

$= 2(\cos(\theta)-1)^{2}\geq 0,$

$\Re(3\ln\zeta(\sigma) + 4\ln\zeta(\sigma+it) + \ln\zeta(\sigma+2it))$

$= \sum_{n=2}^{\infty} \frac{\Lambda(n)}{n^{\sigma}\ln(n)} \cdot ( 3 + 4\cos(t\ln(n)) + \cos(2t\ln(n))) \geq 0.$

$0\leq 3\ln|\zeta(\sigma)| + 4\ln|\zeta(\sigma+it)| + \ln|\zeta(\sigma+2it)|$

$= \ln|\zeta(\sigma)^{3}\zeta(\sigma+it)^{4}\zeta(\sigma+2it)|,$

Claim 3. $\zeta(1+it)\neq 0$ 对于所有的 $\{t\in\mathbb{R}: t\neq 0\}$ 成立。

$\lim_{\sigma\rightarrow 1^{+}} \frac{\zeta(\sigma+it)}{(\sigma+it-1)^{m}}=c\neq 0,$ 其中 $m\geq 1.$

$|(\sigma-1)^{3}\zeta(\sigma)^{3}(\sigma+it-1)^{-4m}\zeta(\sigma+it)^{4}\zeta(\sigma+2it)|$

$\geq |\sigma-1|^{3}|\sigma-1+it|^{-4m}$

$\geq |\sigma-1|^{3}\cdot |\sigma-1|^{-4m}$

$= \frac{1}{|\sigma-1|^{4m-3}}.$

$\sigma\rightarrow 1^{+},$ 可以得到左侧趋近于一个有限的值，但是右侧趋近于无穷，所以得到矛盾。也就是说当 $t\neq 0$ 时， $\zeta(1+it)\neq 0$ 成立。

# 从调和级数到 RIEMANN ZETA 函数（一）

### Riemann Zeta 函数

Riemann Zeta 函数（Riemann zeta function），$\zeta(s)$，是一个关于复数 $s$ 的方程。在复平面上，当复数 $s$ 的实数部分 $\sigma=\Re s >1$ 时，$\zeta(s)$ 就是如下的级数形式：

$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}.$

### 调和级数的概念与性质

$\zeta(1) = \sum_{n=1}^{+\infty}\frac{1}{n}.$

Method 1.

$S_{n}=\sum_{k=1}^{n}\frac{1}{k},$

$|S_{2n}-S_{n}|=\frac{1}{n+1}+...+\frac{1}{2n}>\frac{1}{2n}+...+\frac{1}{2n}=\frac{1}{2},$

Method 2.

$\sum_{n=1}^{+\infty}\frac{1}{n}$

$=1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+...$

$>1+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})+...$

$=1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+...=+\infty.$

Method 3. 调和级数的发散可以通过定积分的技巧来进行解决。

$1+\frac{1}{2}+...+\frac{1}{n}$

$>\int_{1}^{2}\frac{1}{x}dx + \int_{2}^{3}\frac{1}{x}dx+...+\int_{n}^{n+1}\frac{1}{x}dx$

$=\int_{1}^{n+1}\frac{1}{x}dx=\ln(n+1)$

$\lim_{n\rightarrow +\infty}\frac{\sum_{k=1}^{n}\frac{1}{k}}{\ln(n)}$

$= \lim_{n\rightarrow +\infty}\frac{\frac{1}{n}}{\ln(n/(n-1))}$

$= \lim_{x\rightarrow 0}\frac{x}{\ln(1+x)}=1$

$\lim_{n\rightarrow+\infty}(1+\frac{1}{2}+...+\frac{1}{n}-\ln(n))$

### 调和级数的推广

$\zeta(2) = \sum_{n=1}^{\infty}\frac{1}{n^{2}}$

Method 1.

$\sum_{n=1}^{+\infty}\frac{1}{n^{2}}<1+\sum_{n=2}^{+\infty}\frac{1}{n(n-1)}=1+\sum_{n=2}^{+\infty}(\frac{1}{n-1}-\frac{1}{n})=2$.

Method 2. 使用数学归纳法。也就是要证明：

$\sum_{k=1}^{n}1/k^{2}\leq 2-\frac{1}{n}.$

$n=1$ 的时候，公式是正确的。假设 $n$ 的时候是正确的，那么我们有$\sum_{k=1}^{n}1/k^{2}\leq 2-\frac{1}{n}$。计算可得：

$\sum_{k=1}^{n+1}\frac{1}{k^{2}}$

$<2-\frac{1}{n}+\frac{1}{(n+1)^{2}}$

$= 2- \frac{1}{n+1}-\frac{1}{n(n+1)^{2}}$

$\leq 2-\frac{1}{n+1}$.

Method 3.

$1+\frac{1}{2^{2}}+...+\frac{1}{n^{2}}$

$<1+\int_{1}^{2}\frac{1}{x^{2}}dx+...+\int_{n-1}^{n}\frac{1}{x^{2}}dx$

$=1+\int_{1}^{n}\frac{1}{x^{2}}dx=1+1-\frac{1}{n}<2.$

$\zeta(s)=\sum_{n=1}^{+\infty}\frac{1}{n^{s}},$$\sigma = \Re(s)>1.$

### Riemann Zeta 函数中某些点的取值

$a_{0}+\sum_{n=1}^{\infty} (a_{n} \cos(nx) +b_{n} \sin(nx)),$

$a_{n}= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$$n\geq 1,$

$b_{n}= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$$n\geq 1,$

$f(x) =a_{0}+\sum_{n=1}^{\infty} (a_{n} \cos(nx) +b_{n} \sin(nx)).$

$\frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^{2} dx= 2a_{0}^{2}+ \sum_{n=1}^{\infty} (a_{n}^{2}+b_{n}^{2}).$

$\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}}=\frac{\pi^{2}}{8}$

$\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$

$\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{4}}=\frac{\pi^{4}}{96}$

$\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90}$

$\frac{\pi}{2} + \sum_{n=1}^{\infty} \frac{2((-1)^{n}-1)}{\pi} \cdot \frac{cos(nx)}{n^{2}}$

$0= \frac{\pi}{2} + \sum_{n=1}^{\infty} \frac{2((-1)^{n}-1)}{n^{2} \pi} = \frac{\pi}{2} + \sum_{m=1}^{\infty} \frac{-4}{(2m-1)^{2}\pi} = \frac{\pi}{2} - \frac{4}{\pi} \sum_{m=1}^{\infty} \frac{1}{(2m-1)^{2}}$

$S=\sum_{odd} \frac{1}{n^{2}} + \sum_{even} \frac{1}{n^{2}} = \frac{\pi^{2}}{8} + \frac{1}{4} S$.

$\frac{2\pi^{2}}{3}= \frac{1}{\pi} \int_{-\pi}^{\pi} x^{2}dx = 2\cdot (\frac{\pi}{2})^{2} + \sum_{n=1}^{\infty} \frac{4((-1)^{n}-1)^{2}}{\pi^{2}\cdot n^{4}} = \frac{\pi^{2}}{2} + \sum_{m=1}^{\infty} \frac{16}{\pi^{2} (2m-1)^{4}}$

$S=\sum_{odd} \frac{1}{n^{4}} + \sum_{even} \frac{1}{n^{4}} = \frac{\pi^{4}}{96} + \frac{1}{16} S$

# Hausdorff dimension of the graphs of the classical Weierstrass functions

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).

# 从对数学的贡献上来讲，丘成桐有多厉害？

1.丘成桐教授不仅有数学才华，还很有商业天赋。他在Boston地区有三十多套房产。因为Harvard是个很有钱的学校，所以有很多闲置的房产，他们会用极低的价格把这些房产卖给教授。丘成桐教授以其杰出的商业眼光，前前后后一共买了三十多套，租给他的博士后，每年盈利不可胜计，真是令人钦佩！后来丘教授又看中了一处房子，但是学校却不愿意批准卖给他，所以他让当时是系主任的Ben Gross教授去询问缘由，后来Gross说，学校得知你在Boston地区有三十多套房产，实在太多了，所以不能卖给你。大家知道，在数学界，要想组织seminar和conference，经费是必不可少的。正因为丘教授有杰出的商业头脑和投资眼光，所以为中国数学的蓬勃发展输入了大量的物质财富，可谓是中国版的Simons。但是他的数学水平又远胜Simons，所以丘教授无愧为古往今来第一大师！

2.丘教授通过这些seminar和conference让大量的中国年轻数学家有了抛头露面和展示自己的机会。虽然这些年轻人的数学水平只可意会，但是相信通过丘教授的帮助会很快发展成为华人数学界的领军人物，继承他的资源和衣钵。近年来，丘教授在中国大陆，中国香港和台湾地区设立了大量的研究所。这些研究所的设立不但给不少人提供了很好的工作机会，也给不少想学数学的年轻人提供了优秀的平台。比如清华大学的丘成桐数学中心，可以说是亚洲第一数学中心，连日本京都的RIMS都是远远不如的，我想即使放到宇宙上也是名列前茅的。在这里我们应该特别欢迎广大二本和三本的数学系学生报考这些研究所，因为丘先生的理念就是要给普通高校热爱数学的学生以机会。

3.丘教授每年都到中国的各所高校讲学，尤其是他开设的几个数学中心，这些讲座传授给年轻人许多高深的数学知识和实用的数学技巧。他演讲的话题包括：数学之美、我的成功经验、Harvard数学系的历史和我的一个不听话的学生等等。内容丰富，发人深省，不但能从中学到数学知识，还能体会到许多做（中国）人的道理。可悲的是，一些反动派受到西方自由思想的荼毒，对这样高质量的讲座却视而不见，拒绝参加，其中包括一些数学界的同行。丘教授知悉此事后，给这些人发了一封邮件，明确要求他们：今后只要是我来你们学校做讲座，所有中国人就必须参加！丘先生的严厉做法很好地整肃了华人数学界的风气，提高了凝聚力。相信在丘先生的领导下，大家一定能鼓足干劲，力争上游，多快好省地建设中国数学！

4.丘教授亲自培养的许多学生都有极高的数学水准，在国际上获得广泛承认，多次荣获重大国际奖项，比如晨兴数学奖、新世界数学奖、陈省身奖之中国版等等。这些学生不仅自己水平惊人，对年轻人也提供了无微不至的关怀和细致周到的帮助。比如，丘教授的不少学生害怕学生没有自己的想法，经常亲自给学生提供idea，来帮助学生找到研究的思路。即使学生不需要也要苦口薄心，再三敦促。这样一来，不仅学生可以发paper，他们自己也因为贡献了一个“关键的”idea而顺便加到了名字，可谓是一举两得的做法。丘教授另一些学生因为害怕国际上一些著名杂志的编辑是势利眼，不让年轻学生单独发paper，所以不惜牺牲自己的名节，主动要求在paper上加名字。这样一来，学生发文章的时候就不会吃亏了。他们为学生的付出令人感动。可悲的是，一些年轻人不但不知道感恩，反而对此感到苦恼。对这样的人，我们就应该毫不犹豫地把他们踢出华人数学界，让他们去落后的西方世界吃点苦头！

5.丘教授掌握了国际上一本极为重要的数学杂志，即Journal of Differential Geometry。这本杂志现在成为许多年轻人展示自己只可意会的数学水平和找到教职的最佳平台。为了方便某些中国学生在杂志上发表论文，丘教授提供了一些非同寻常的便捷渠道。比如文章不用发给编辑，可以直接发给自己，再由他转发给编辑。这样一来，中国数学家的文章就经常出现在顶级杂志上，他们的研究水准得到了空前飞跃！丘教授控制的另一本杂志就是大名鼎鼎的Asian Journal。这本杂志上发表了人类在20世纪到21世纪一些最伟大的数学工作，比如朱熹平教授和曹怀东教授对Poincare猜想的最终证明，封顶了人类一百余年来悬而未决的难题。这篇文章长达300多页，但是经过Asian Journal的编辑不知疲倦的辛勤工作，该论文在极短的时间内就获得了发表。可以看到，丘教授在经营杂志以后，杂志审核文章的效率大大提高了。可以说，正是丘教授勤劳刻苦，生命不息，奋斗不止的精神感召了这些编辑，让他们不再玩忽职守和放松懈怠。

6.丘成桐教授对自己学生的关怀可以说是无微不至。有些学生一时糊涂涉嫌抄袭和剽窃，丘教授知道以后果断采取措施，息事宁人，避免了家丑外扬。中国数学界正是在丘先生的努力下才能铁板一块地团结在一起，大家毫无私心，全心全意为中国数学的发展添砖加瓦。但是有些人却不明白丘教授的苦心，经常在丘教授面前投诉，甚至还写匿名信把事情闹到别的学校。对此，丘教授态度坚决，铁面无私地无视了这些无理要求，可以说很好地体现了一位领袖的英明果决。而那些闹事的逆流虽然可能有一点点数学水平，但是今天也没办法站出来领导数学界了。就是因为某些人只知道做研究和思考数学问题，没有意识到帮助中国数学发展才是更有意义的事。思想境界比起丘教授差的太远了。可以说，丘先生高瞻远瞩，气盖环宇，数风流人物，还看今朝。

7.丘教授对中国学生的关心不仅仅局限在数学系，还遍及到各个非数学领域。从前，只要是中国、香港和台湾去Harvard读数学的学生，丘教授都要亲自过问，热情关怀，把他们一一纳入自己门下。比如某学生要跟Taubes，他会亲自找到Taubes，告诉他，这位学生就托付给你了。这样一来，这些西方数学家慑于丘先生的气魄和威望，就不敢再歧视中国学生了。到了后来，只要去Harvard的中国、香港和台湾学生，无论学什么专业，丘先生都要跟他们打交道。据说他还曾经举办过大型party，邀请Harvard商学院大中华地区的所有学生参加。这些活动使他亲民的形象更加突出，在各界广受好评。相信不久的将来，丘教授会吸引到亚洲其他地区的学生参与他的party。像他这的一代王者，相信任何人都会被他的魅力所感召。毕竟只有深入到人民群众中去，才能发现问题所在。丘教授真不愧为一代明君！

8.丘教授虽然已经接近70高龄，仍然老骥伏枥，近年来在数学研究上非常活跃。仅2015一年就在arxiv贴文23篇，以每个月两篇论文的速度进行高质量的数学研究，这是古往今来其他任何数学家都望尘莫及的！要知道，丘教授作为华人数学界的领袖，每天要处理几百封邮件。熟悉丘教授的朋友们都知道，即使是在seminar上他也要一边摁手机收发邮件，一边听talk。能在如此繁忙的情况下一个月写两篇论文，效率之高真是令人震惊！丘教授还特别注意与年轻人的合作，近年来每篇论文几乎都要提携一些年轻数学家，大度地和他们一起署名发表。由于他提携的年轻数学家太多，很多时候甚至会忘记自己的合作者。比如某韩国数学家之前跟他有合作，到了找教职的时候希望丘教授能帮自己写推荐信，但是丘教授却坦言自己并不认识对方。实际上，丘教授不认识自己的合作者正可以反映出他已经帮助了太多年轻人，以至于自己都想不起来自己干的那些好事！范仲淹说：云山苍苍，江水泱泱，先生之风，山高水长。丘先生年近七旬而笔耕不辍，真可谓吾辈典范！

9.丘成桐教授对于人才优劣的判断也是明察秋毫，一望即知。早先，北大一个学生仗着自己是那一届最优秀的就自不量力，想要去Harvard跟丘教授学数学，丘教授对他说：你水平不行。想跟我也可以，先去Boston待两年，经我考察合格了，再来跟我。这个学生不得已之下去了另一个inferior的学校跟了一个比丘教授差了十万八千里的数学家M。事实证明，这个学生现在虽然出了一点小名，在Yale做教授，但是确实不够资格在Harvard做丘教授的学生：因为他只拿到了晨兴数学银奖，而丘教授的学生一般都是拿金奖的。

——————————————–

# 低维动力系统

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

### 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

# Complex Analysis

（1）提到复变函数，首先需要了解复数 (Complex Numbers) 的基本性质和四则运算规则。怎么样计算复数的平方根，极坐标与xy坐标的转换，复数的模之类的。这些在高中的时候基本上都会学过。

（2）复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之类的运算就会很自然的引入到复平面里面，从而引出解析函数 (Holomorphic Functions / Analytic Functions) 的定义。那么研究解析函数的性质就是关键所在。最关键的地方就是所谓的Cauchy—Riemann公式，这个是判断一个函数是否是解析函数的关键所在。

（3）明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 (Line Integrals) 的概念引入复分析中，定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理：Cauchy积分公式 (Cauchy’s Integral Formula)。这个是复分析的第一个重要定理。

（4）既然是解析函数，那么函数的定义域 (Domain) 就是一个关键的问题。可以从整个定义域去考虑这个函数，也可以从局部来研究这个函数。这个时候研究解析函数的奇点 (Singularity) 就是关键所在，奇点根据性质分成可去奇点 (Removable Singularity)，极点 (Pole)，本性奇点 (Essential Singularity) 三类，围绕这三类奇点，会有各自奇妙的定理。

（5）复变函数中，留数定理 (Residue Theorem) 是一个重要的定理，反映了曲线积分和零点极点的性质。与之类似的幅角定理也展示了类似的关系。

（6）除了积分，导数也是解析函数的一个研究方向。导数加上收敛 (Convergence) 的概念就可以引出 Taylor 级数 (Taylor Series) 和 Laurent 级数 (Laurent Series) 的概念。除此之外，正规族 (Normal Families) 里面有一个非常重要的定理，那就是Arzela定理。

（7）以上都是从分析的角度来研究复分析，如果从几何的角度来说，最重要的定理莫过于 Riemann 映照定理 (Riemann Mapping Theorem)。这个时候一般会介绍线性变换，就是 Mobius 变换 (Mobius Transforms)，把各种各样的单连通区域映射成单位圆。研究 Mobius 变换的保角和交比之类的性质。

（8）椭圆函数 (Elliptic Functions)，经典的双周期函数 (Double Periodic Functions)。这里有 Weierstrass 理论，是研究 Weierstrass 函数的，有经典的微分方程，以及该函数的性质。 以上就是复分析或者复变函数的一些课程介绍，如果有遗漏或者疏忽的地方请大家指教。

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

（2）Complex Analysis，Elias M. Stein

# 调和分析

### (6) Ap weight

ps：这是2009年的事情了，一晃眼7年过去了。

Loukas Grafakos GTM249 Classical Fourier Analysis
Loukas Grafakos GTM250 Modern Fourier Analysis
(上面这两本书是调和分析的经典之作，几乎涵盖了实变方法的所有内容。不过有点厚，差不多1100页。)

# Ergodic Properties

## 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 $f^{m}(x)\in U$.

# Fractals – A Very Short Introduction

Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.

# Chapter 7 A little history

Geometry, with its highly visual and practical nature, is one of the oldest branches of mathematics. Its development through the ages has paralleled its increasingly sophisticated applications. Construction, crafts, and astronomy practised by ancient civilizations led to the need to record and analyse the shapes, sizes, and positions of objects. Notions of angles, areas, and volumes developed with the need for surveying and building. Two shapes were especially important: the straight line and the circle, which occurred naturally in many settings but also underlay the design of many artefacts. As well as fulfilling practical needs, philosophers were motivated by aesthetic aspects of geometry and sought simplicity in geometric structures and their applications. This reached its peak with the Greek School, notably with Plato (c 428–348 BC) and Euclid (c 325–265 BC), for whom constructions using a straight edge and compass, corresponding to line and circle, were the essence of geometric perfection.

As time progressed, ways were found to express and solve geometrical problems using algebra. A major advance was the introduction by René Descartes (1596–1650) of the Cartesian coordinate system which enabled shapes to be expressed concisely in terms of equations. This was a necessary precursor to the calculus, developed independently by Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1714) in the late 17th century. The calculus provided a mathematical procedure for finding tangent lines that touched smooth curves as well as a method for computing areas and volumes of an enormous variety of geometrical objects. Alongside this, more sophisticated geometric figures were being observed in nature and explained mathematically. For example, using Tycho Brahe’s observations, Johannes Kepler proposed that planets moved around ellipses, and this was substantiated as a mathematical consequence of Newton’s laws of motion and gravitation.

The tools and methods were now available for tremendous advances in mathematics and the sciences. All manner of geometrical shapes could be analysed. Using the laws of motion together with the calculus, one could calculate the trajectories of projectiles, the motion of celestial bodies, and, using differential equations which developed from the calculus, more complex motions such as fluid flows. Although the calculus underlay Graph of a Brownian process8I to think of all these applications, its foundations remained intuitive rather than rigorous until the 19th century when a number of leading mathematicians including Augustin Cauchy (1789–1857), Bernhard Riemann (1826–66), and Karl Weierstrass (1815–97) formalized the notions of continuity and limits. In particular, they developed a precise definition for a curve to be ‘differentiable’, that is for there to be a tangent line touching the curve at a point. Many mathematicians worked on the assumption that all curves worthy of attention were nice and smooth so had tangents at all their points, enabling application of the calculus and its many consequences. It was a surprise when, in 1872, Karl Weierstrass constructed a ‘curve’ that was so irregular that at no point at all was it possible to draw a tangent line. The Weierstrass graph might be regarded as the first formally defined fractal, and indeed it has been shown to have fractal dimension greater than 1.

In 1883, the German Georg Cantor (1845–1918) wrote a paper introducing the middle-third Cantor set, obtained by repeatedly removing the middle thirds of intervals (see Figure 44). The Cantor set is perhaps the most basic self-similar fractal, made up of 2 scale copies of itself, although of more immediate interest to Cantor were its topological and set theoretic properties, such as it being totally disconnected, rather than its geometry. (Several other mathematicians studied sets of a similar form around the same time, including the Oxford mathematician Henry Smith (1826–83) in an article in 1874.) In 1904, Helge von Koch introduced his curve, as a simpler construction than Weierstrass’s example of a curve without any tangents. Then, in 1915, the Polish mathematician Wacław Sierpiński (1882–1969) introduced his triangle and, in 1916, the Sierpiński carpet. His main interest in the carpet was that it was a ‘universal’ set, in that it contains continuously deformed copies of all sets of ‘topological dimension’ 1. Although such objects have in recent years become the best-known fractals, at the time properties such as self-similarity were almost irrelevant, their main use being to provide specific examples or counter-examples in topology and calculus.

It was in 1918 that Felix Hausdorff proposed a natural way of ‘measuring’ the middle-third Cantor set and related sets, utilizing a general approach due to Constantin Carathéodory (1873–1950). Hausdorff showed that the middle-third Cantor set had dimension of log2/log3 = 0.631, and also found the dimensions of other self-similar sets. This was the first occurrence of an explicit notion of fractional dimension. Now termed ‘Hausdorff dimension’, his definition of dimension is the one most commonly used by mathematicians today. (Hausdorff, who did foundational work in several other areas of mathematics and philosophy, was a German Jew who tragically committed suicide in 1942 to avoid being sent to a concentration camp.) Box-dimension, which in many ways is rather simpler than Hausdorff dimension, appeared in a 1928 paper by Georges Bouligand (1889–1979), though the idea underlying an equivalent definition had been mentioned rather earlier by Hermann Minkowski (1864–1909), a Polish mathematician known especially for his work on relativity.

For many years, few mathematicians were very interested in fractional dimensions, with highly irregular sets continuing to be regarded as pathological curiosities. One notable exception was Abram Besicovitch (1891–1970), a Russian mathematician who held a professorship in Cambridge for many years. He, along with a few pupils, investigated the dimension of a range of fractals as well as investigating some of their geometric properties.

Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.