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？

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• 理论知识太多；
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• 底层通用技能；
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• 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。

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• 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.

Perron-Frobenius Operator

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} or $q_{0}

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 $f(I_{k})\supseteq I_{j}$;

(c) there exists $r>0$ such that $|f(I_{i})|\geq r$ for each $i$.

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;

(c) $f$ is exact with respect to $\mu$;

(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.

Shape of Inner Space

String Theory and the Geometry of the Universe’s Hidden Dimensions

Chapter 3: P.39

My personal involvement in this area began in 1969, during my first semester of graduate studies at Berkeley. I needed a book to read during Chrismas break. Rather than selecting Portnoy’s Complaint, The Godfather, The Love Machine, or The Andromeda Strain-four top-selling books of that year-I opted for a less popular title, Morse Theory, by the American mathematician John Milnor. I was especially intrigued by Milnor’s section on topology and curvature, which explored the notion that local curvature has a great influence on geometry and topology. This is a theme I’ve pursued ever since, because the local curvature of a surface is determined by taking the derivatives of that surface, which is another way of saying it is based on analysis. Studying how that curvature influences geometry, therefore, goes to the heart of geometric analysis.

Having no office, I practically lived in Berkeley’s math library in those days. Rumor has it that the first thing I did upon arriving in the United States was visiting that library, rather than, say, explore San Francisco as other might have done. While I can’t remember exactly what I did, forty years hence, I have no reason to doubt the veracity of that rumor. I wandered around the library, as was my habit, reading every journal I could get my hands on. In the course of rummaging through the reference section during winter break, I came across a 1968 article by Milnor, whose book I was still reading. That article, in turn, little else to do at the time (with most people away for the holiday), I tried to see if I could prove something related to Preissman’s theorem.

Chapter 4: P.80

From this sprang the work I’ve become most famous for. One might say it was my calling. No matter what our station, we’d all like to find our true calling in life-that special thing we were put on this earth to do. For an actor, it might be playing Stanley Kowalski in A Streetcar Named Desire. Or the lead role in Hamlet. For a firefighter, it could mean putting out a ten-alarm blaze. For a crime-fighter, it could mean capturing Public Enemy Number One. And in mathematics, it might come down to finding that one problem you’re destined to work on. Or maybe destiny has nothing to do with it. Maybe it’s just a question of finding a problem you can get lucky with.

To be perfectly honest, I never think about “destiny” when choosing a problem to work on, as I tend to be a bit more pragmatic. I try to seek out a new direction that could bring to light new mathematical problems, some of which might prove interesting in themselves. Or I might pick an existing problem that offers the hope that in the course of trying to understand it better, we will be led to a new horizon.

The Calabi conjecture, having been around a couple of decades, fell into the latter category. I latched on to this problem during my first year of graduate school, though sometimes it seemed as if the problem latched on to me. It caught my interest in a way that no other problem had before or has since, as I sensed that it could open a door to a new branch of mathematics. While the conjecture was vaguely related to Poincare’s classic problem, it struck me as more general because if Calabi’s hunch were true, it would lead to a large class of mathematical surfaces and spaces that we didn’t know anything about-and perhaps a new understanding of space-time. For me the conjecture was almost inescapable: Just about every road I pursued in my early investigations of curvature led to it.

Chapter 5: P.104

A mathematical proof is a bit like climbing a mountain. The first stage, of course, is discovering a mountain worth climbing. Imagine a remote wilderness area yet to be explored. It takes some wit just to find such an area, let alone to know whether something worthwhile might be found there. The mountaineer then devises a strategy for getting to the top-a plan that appears flawless, at least on paper. After acquiring the necessary tools and equipment, as well as mastering the necessary skills, the adventurer mounts an ascent, only to be stopped by unexpected difficulties. But others follow in their predecessor’s footsteps, using the successful strategies, while also pursuing different avenues-thereby reaching new heights in the process. Finally someone comes along who not only has a good plan of attack that avoids the pitfalls of the past but also has the fortitude and determination to reach the summit, perhaps planting a flag there to mark his or her presence. The risks to life and limb are not so great in math, and the adventure may not be so apparent to the outsider. And at the end of a long proof, the scholar does not plant a flag. He or she types in a period. Or a footnote. Or a technical appendix. Nevertheless, in our field there are thrill as well as perils to be had in the pursuit, and success still rewards those of us who’ve gained new views into nature’s hidden recesses.

Normal Families

Reference Book: Joel L.Schiff- Normal Families

Some Classical Theorems

Weierstrass Theorem Let $\{ f_{n}\}$ be a sequence of analytic functions on a domain $\Omega$ which converges uniformly on compact subsets of $\Omega$ to a function $f$. Then $f$ is analytic in $\Omega$, and the sequence of derivatives $\{ f_{n}^{(k)}\}$ converges uniformly on compact subsets to $f^{(k)}, k=1,2,3...$.

Hurwitz Theorem Let $\{ f_{n}\}$ be a sequence of analytic functions on a domain $\Omega$ which converges uniformly on compact subsets of $\Omega$ to a non-constant analytic function $f(z)$. If $f(z_{0})=0$ for some $z_{0}\in\Omega$, then for each $r>0$ sufficiently small, there exists an $N=N(r)$, such that for all $n>N$, $f_{n}(z)$ has the same number of zeros in $D(z_{0},r)$ as does $f(z)$. (The zeros are counted according to multiplicity).

The Maximum Principle If $f(z)$ is analytic and non-constant in a region $\Omega$, then its absolute value $|f(z)|$ has no maximum in $\Omega$.

The Maximum Principle’ If $f(z)$ is defined and continuous on a closed bounded set $E$ and analytic on the interior of $E$, then the maximum of $|f(z)|$ on $E$ is assumed on the boundary of $E$.

Corollary 1.4.1 If $\{ f_{n}\}$ is a sequence of univalent analytic functions in a domain $\Omega$ which converge uniformly on compact subsets of $\Omega$ to a non-constant analytic function $f$, then $f$ is univalent in $\Omega$.

Definition 1.5.1 A family of functions $\mathcal{F}$ is locally bounded on a domain $\Omega$ if, for each $z_{0}\in \Omega$, there is a positive number $M=M(z_{0})$ and a neighbourhood $D(z_{0},r)\subset \Omega$ such that $|f(z)|\leq M$ for all $z\in D(z_{0}, r)$ and all $f\in \mathcal{F}$.

Theorem 1.5.2 If $\mathcal{F}$ is a family of locally bounded analytic functions on a domain $\Omega$, then the family of derivatives $\mathcal{F}^{'}=\{ f^{'}: f\in \mathcal{F}\}$ form a locally bounded family in $\Omega$.

The converse of Theorem 1.5.2 is false, since $\mathcal{F}=\{n: n=1,2,3...\}$. However, the following partial converse does hold.

Theorem 1.5.3 Let $\mathcal{F}$ be a family of analytic functions on $\Omega$ such that the family of derivatives $\mathcal{F}^{'}$ is locally bounded and suppose that there is some $z_{0}\in \Omega$ with $|f(z_{0})|\leq M<\infty$ for all $f\in \mathcal{F}$. Then $\mathcal{F}$ is locally bounded. (Hint: find a path connecting $z_{0}$ and $z$.)

Definition 1.6.1 A family $\mathcal{F}$ of functions defined on a domain $\Omega$ is said to be equicontinuous (spherically continuous) at a point $z^{'}\in \Omega$ if, for each $\epsilon>0$, there is a $\delta=\delta(\epsilon,z^{'})>0$ such that $|f(z)-f(z^{'})|<\epsilon$$(\chi(f(z),f(z^{'}))<\epsilon)$ whenever $|z-z^{'}|<\delta$, for every $f\in \mathcal{F}$. Moreover, $\mathcal{F}$ is equicontinuous (spherical continuous) on a subset $E\subset \Omega$ if it is continuous (spherically continuous) at each point of $E$.

Normal Families of Analytic Functions

Definition 2.1.1  A familiy $\mathcal{F}$ of  analytic functions on a domain $\Omega\subset \mathbb{C}$ is normal in $\Omega$ if every sequence of functions $\{f_{n}\}\subset \mathcal{F}$ contains either a subsequence which converges to a limit function $f\not\equiv \infty$ uniformly on each compact subset of $\Omega$, or a subsequence which converges uniformly to $\infty$ on each compact subset.

The family $\mathcal{F}$ is said to be normal at a point $z_{0}\in\Omega$ if it is normal in some neighbourhood of $z_{0}$.

Theorem 2.1.2 A family of analytic functions $\mathcal{F}$ is normal in a domain $\Omega$ if and only if $\mathcal{F}$ is normal at each point in $\Omega$.

Theorem 2.2.1 Arzela-Ascoli Theorem. If a sequence $\{f_{n}\}$ of continuous functions converges uniformly on a compact set $K$ to a limit function $f\not\equiv \infty$, then $\{f_{n}\}$ is equicontinuous on $K$, and $f$ is continuous. Conversely, if $\{f_{n}\}$ is equicontinuous and locally bounded on $\Omega$, then a subsequence can be extracted from $\{f_{n}\}$ which converges locally uniformly in $\Omega$ to a (continuous) limit function $f$.

Montel’s Theorem If $\mathcal{F}$ is a locally bounded family of analytic functions on a domain $\Omega$, then $\mathcal{F}$ is a normal family in $\Omega$.

Koebe Distortion Theorem Let $f(z)$ be analytic univalent in a domain $\Omega$ and $K$ a compact subset of $\Omega$. Then there exists a constant $c=c(\Omega, K)$ such that for any $z,w\in K$, $c^{-1}\leq |f^{'}(z)| / |f^{'}(w)| \leq c$.

Vitali-Porter Theorem Let $\{f_{n}\}$ be a locally bounded sequence of analytic functions in a domain $\Omega$ such that $\lim_{n\rightarrow \infty}f_{n}(z)$ exists for each $z$ belonging to a set $E\subset \Omega$ which has an accumulation point in $\Omega$. Then $\{ f_{n}\}$ converges uniformly on compact subsets of $\Omega$ to an analytic function.

Proof. From Montel’s Theorem, $\{ f_{n}\}$ is normal, extract a subsequence $\{ f_{n_{k}}\}$ which converges normally to an analytic function $f$. Then $\lim_{k\rightarrow \infty} f_{n_{k}}(z)=f(z)$ for each $z\in E$.  Suppose, however, that $\{ f_{n}\}$ does not converge uniformly on compact subsets of $\Omega$ to $f$. Then there exists some $\epsilon>0$, a compact subset $K\subset \Omega$, as well as a subsequence $\{f_{m_{j}}\}$ and points $z_{j}\in K$ satisfying $|f_{m_{j}}(z_{j})- f(z_{j})| \geq \epsilon,$ $j=1,2,3,...$. Now $\{ f_{m_{j}}\}$ itself has a subsequence which converges uniformly on compact subsets to an analytic function $g$, and $g\not\equiv f$ from above. However, since $f$ and $g$ must agree at all points of $E$, the Identity Theorem for analytic functions implies $f\equiv g$ on $\Omega$, a contradiction which establishes the theorem.

Fundamental Normality Test Let $\mathcal{F}$ be the family of analytic functions on a domain $\Omega$ which omit two fixed values $a$ and $b$ in $\mathbb{C}$. Then $\mathcal{F}$ is normal in $\Omega$.

Generalized Normality Test Suppose that $\mathcal{F}$ is a family of analytic functions in a domain $\Omega$ which omit a value $a\in \mathbb{C}$ and such that no function of $\mathcal{F}$ assumes the value $b\in \mathbb{C}$ at more that $p$ points. Then $\mathcal{F}$ is normal in $\Omega$.

2.3 Examples:

Assume $U$ is the unit disk in the complex plane, $\Omega$ is a region (connected open set) in $\mathbb{C}$.

1. $\mathcal{F}=\{ f_{n}(z)=z^{n}: n=1,2,3...\}$ in $U$. Then $\mathcal{F}$ is normal in $U$, but not compact since $0 \notin \mathcal{F}$. In the domain $U^{'}: |z|>1$, $\mathcal{F}$ is normal.

2. $\mathcal{F}=\{ f_{n}(z)=\frac{z}{n}: n=1,2,3...\}$ is a normal family in $\mathcal{C}$ but not compact.

3. $\mathcal{F}=\{ f: f$ analytic in $\Omega$  and $|f|\leq M \}$. Then $\mathcal{F}$ is normal in $\Omega$ and compact.

4. $\mathcal{F}=\{ f: f$ analytic in $\Omega$ and $\Re f>0\}$. Then $\mathcal{F}$ is normal but not compact. Hint: $\mathcal{G}=\{g=e^{-f}:f\in \mathcal{F}\}$ is a uniformly bounded family.

5. $\mathcal{S}=\{ f: f$ analytic, univalent in $U$, $f(0)=0, f^{'}(0)=1 \}$. These are the normalised “Schlicht” functions in $U$. $\mathcal{S}$ is normal and compact.

Normal Families of Meromorphic Functions

Assume a function $f(z)$ is analytic in a neighbourhood of $a$, except perhaps at $a$ itself. In other words, $f(z)$ shall be analytic in a region $0<|z-a|<\delta$. The point $a$ is called an isolated singularity of $f(z)$. There are three cases about an isolated singularity. The first one is a removable singularity, the second one is a pole, the third one is an essential singularity.  A function $f(z)$ which is analytic in a region $\Omega$, except for poles, is said to be meromorphic in $\Omega$.

The chordal distance $\chi(z_{1}, z_{2})$ between $z_{1}$ and $z_{2}$ is

$\chi(z_{1}, z_{2}) = \frac{|z_{1}-z_{2}|}{\sqrt{1+|z_{1}|^{2}}\sqrt{1+|z_{2}|^{2}}}$ if $z_{1}$ and $z_{2}$ are in the finite plane, and

$\chi(z_{1}, \infty) = \frac{1}{\sqrt{1+|z_{1}|^{2}}},$ if $z_{2}=\infty$. Clearly, $\chi(z_{1}, z_{2})\leq 1$, and $\chi(z_{1}^{-1}, z_{2}^{-1}) = \chi(z_{1}, z_{2})$. The chordal metric and spherical metric are uniformly equivalent and generate the same open sets on the Riemann sphere.

Definition 1.2.1 A sequence of functions $\{ f_{n}\}$ converges spherically uniformly to $f$ on a set $E\subset \mathbb{C}$ if, for any $\epsilon>0$, there is a number $n_{0}$ such that $n\geq n_{0}$ implies $\chi(f(z), f_{n}(z))<\epsilon$, for all $z\in E$.

Definition 3.1.1 A family $\mathcal{F}$ of meromorphic functions in a domain $\Omega$ is normal in $\Omega$ if every sequence $\{ f_{n} \} \subset \mathcal{F}$ contains a subsequence which converges spherically uniformly on compact subsets of $\Omega$.

Theorem 3.1.3 Let $\{ f_{n}\}$ be a sequence of meromorphic functions on a domain $\Omega$. Then $\{ f_{n}\}$ converges spherically uniformly on compact subsets of $\Omega$ to $f$ if and only if about each point $z_{0}\in \Omega$ there is a closed disk $K(z_{0},r)$ in which $|f_{n}-f|\rightarrow 0$ or $|1/f_{n} - 1/f| \rightarrow 0$ uniformly as $n\rightarrow \infty$.

Corollary 3.1.4 Let $\{ f_{n}\}$ be a sequence of meromorphic functions on $\Omega$ which converges spherically uniformly on compact subsets to $f$. Then $f$ is either a meromorphic function on $\Omega$ or identically equal to $\infty$.

Corollary 3.1.5  Let $\{ f_{n}\}$ be a sequence of analytic functions on a domain $\Omega$ which converge spherically uniformly on compact subsets of $\Omega$ to $f$. Then $f$ is either analytic on $\Omega$ or identically equal to $\infty$.

Theorem 3.2.1 A family $\mathcal{F}$ of meromorphic functions in a domain $\Omega$ is normal if and only if $\mathcal{F}$ is spherically equicontinuous in $\Omega$.

Fundamental Normality Test Let $\mathcal{F}$ be a family of meromorphic functions on a domain $\Omega$ which omit three distinct values $a, b, c \in \mathbb{C}$. Then $\mathcal{F}$ is normal in $\Omega$.

Vitali-Porter Theorem Let $\{f_{n}\}$ be a sequence belonging to a spherically equicontinuous family of meromorphic functions such that $\{ f_{n}(z)\}$ converges spherically on a point set $E$ having an accumulation point in $\Omega$. Then $\{ f_{n}\}$ converges spherically uniformly on compact subsets of $\Omega$.

Let $f(z)$ be meromorphic on a domain $\Omega$. If $z\in \Omega$ is not a pole, the derivative in the spherical metric, called the spherical derivative, is given by $f^{\#}(z) =\lim_{z^{'}\rightarrow z}\frac{\chi(f(z),f(z^{'}))}{|z-z^{'}|} =\frac{|f^{'}(z) |}{1+|f(z)|^{2}}$. If $\zeta$ is a pole of $f(z)$, define $f^{\#}(\zeta) = \lim_{z\rightarrow \zeta} \frac{|f^{'}(z)|}{1+|f(z)|^{2}}$.

Marty’s Theorem A family $\mathcal{F}$ of meromorphic functions on a domain $\Omega$ is normal if and only if for each compact subset $K\subset \Omega$, there exists a constant $C=C(K)$ such that spherical derivative $f^{\#}(z) =\frac{|f^{'}(z) |}{1+|f(z)|^{2}}\leq C, z\in K, f\in \mathcal{F},$ that is, $f^{\#}$ is locally bounded.

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2014 International Congress of Mathematics: Awards

Fields Medalist:

Artur Avila

CNRS, France & IMPA, Brazil

[Artur Avila is awarded a Fields Medal] for his profound contributions to dynamical systems theory have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

Avila leads and shapes the field of dynamical systems. With his collaborators, he has made essential progress in many areas, including real and complex one-dimensional dynamics, spectral theory of the one-frequency Schródinger operator, flat billiards and partially hyperbolic dynamics.

Avila’s work on real one-dimensional dynamics brought completion to the subject, with full understanding of the probabilistic point of view, accompanied by a complete renormalization theory. His work in complex dynamics led to a thorough understanding of the fractal geometry of Feigenbaum Julia sets.

In the spectral theory of one-frequency difference Schródinger operators, Avila came up with a global de- scription of the phase transitions between discrete and absolutely continuous spectra, establishing surprising stratified analyticity of the Lyapunov exponent.

In the theory of flat billiards, Avila proved several long-standing conjectures on the ergodic behavior of interval-exchange maps. He made deep advances in our understanding of the stable ergodicity of typical partially hyperbolic systems.

Avila’s collaborative approach is an inspiration for a new generation of mathematicians.

Manjul Bhargava

Princeton University, USA

[Manjul Bhargava is awarded a Fields Medal] for developing powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves.

Bhargava’s thesis provided a reformulation of Gauss’s law for the composition of two binary quadratic forms. He showed that the orbits of the group SL(2, Z)3 on the tensor product of three copies of the standard integral representation correspond to quadratic rings (rings of rank 2 over Z) together with three ideal classes whose product is trivial. This recovers Gauss’s composition law in an original and computationally effective manner. He then studied orbits in more complicated integral representations, which correspond to cubic, quartic, and quintic rings, and counted the number of such rings with bounded discriminant.

Bhargava next turned to the study of representations with a polynomial ring of invariants. The simplest such representation is given by the action of PGL(2, Z) on the space of binary quartic forms. This has two independent invariants, which are related to the moduli of elliptic curves. Together with his student Arul Shankar, Bhargava used delicate estimates on the number of integral orbits of bounded height to bound the average rank of elliptic curves. Generalizing these methods to curves of higher genus, he recently showed that most hyperelliptic curves of genus at least two have no rational points.

Bhargava’s work is based both on a deep understanding of the representations of arithmetic groups and a unique blend of algebraic and analytic expertise.

Martin Hairer

University of Warwick, UK

[Martin Hairer is awarded a Fields Medal] for his outstanding contributions to the theory of stochastic partial differential equations, and in particular created a theory of regularity structures for such equations.

A mathematical  problem that  is important  throughout science is to understand the influence of noise on differential equations, and on the long time behavior of the solutions. This problem was solved for ordinary differential equations by Itó in the 1940s. For partial differential equations, a comprehensive theory has proved to be more elusive, and only particular cases (linear equations, tame nonlinearities, etc.)  had been treated satisfactorily.

Hairer’s work addresses two central aspects of the theory.  Together with Mattingly  he employed the Malliavin calculus along with new methods to establish the ergodicity of the two-dimensional stochastic Navier-Stokes equation.

Building  on the rough-path approach of Lyons for stochastic ordinary differential equations, Hairer then created an abstract theory of regularity structures for stochastic partial differential equations (SPDEs). This allows Taylor-like expansions around any point in space and time. The new theory allowed him to construct systematically solutions to singular non-linear SPDEs  as fixed points of a renormalization procedure.

Hairer was thus able to give, for the first time, a rigorous intrinsic meaning to many SPDEs arising in physics.

Maryam Mirzakhani

Stanford University, USA

[Maryam Mirzakhani is awarded the Fields Medal] for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

Maryam Mirzakhani has made stunning advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new frontiers in this area. Her insights have integrated methods from diverse fields, such as algebraic geometry, topology and probability theory.

In hyperbolic geometry, Mirzakhani established asymptotic formulas and statistics for the number of simple closed geodesics on a Riemann surface of genus g. She next used these results to give a new and completely unexpected proof of Witten’s conjecture, a formula for characteristic classes for the moduli spaces of Riemann surfaces with marked points.

In dynamics, she found a remarkable new construction that bridges the holomorphic and symplectic aspects of moduli space, and used it to show that Thurston’s earthquake flow is ergodic and mixing.

Most recently, in the complex realm, Mirzakhani and her coworkers produced the long sought-after proof of the conjecture that – while the closure of a real geodesic in moduli space can be a fractal cobweb, defying classification – the closure of a complex geodesic is always an algebraic subvariety.

Her work has revealed that the rigidity theory of homogeneous spaces (developed by Margulis, Ratner and others) has a definite resonance in the highly inhomogeneous, but equally fundamental realm of moduli spaces, where many developments are still unfolding

Nevanlinna Prize Winner:

Subhash Khot

New York University, USA

[Subhash Khot is awarded the Nevanlinna Prize] for his prescient  definition of the “Unique Games” problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.

Subhash Khot defined the “Unique Games” in 2002 , and subsequently led the effort to understand its complexity and its pivotal role in the study of optimization problems. Khot and his collaborators demonstrated that the hardness of Unique Games implies a precise characterization of the best approximation factors achievable for a variety of NP-hard optimization problems. This discovery turned the Unique Games problem into a major open problem of the theory of computation.

The ongoing quest to study its complexity has had unexpected benefits. First, the reductions used in the above results identified new problems in analysis and geometry, invigorating analysis of Boolean functions, a field at the interface of mathematics and computer science. This led to new central limit theorems, invariance principles, isoperimetric inequalities, and inverse theorems, impacting research in computational complexity, pseudorandomness, learning and combinatorics. Second, Khot and his collaborators used intuitions stemming from their study of Unique Games to yield new lower bounds on the distortion incurred when embedding one metric space into another, as well as constructions of hard families of instances for common linear and semi- definite programming algorithms. This has inspired new work in algorithm design extending these methods, greatly enriching the theory of algorithms and its applications.

Gauss Prize Winner:

Stanley Osher

University of Califonia, USA

[Stanley Osher is awarded the Gauss Prize] for his influential contributions to several fields in applied mathematics, and his far-ranging inventions have changed our conception of physical, perceptual, and mathematical concepts, giving us new tools to apprehend the world.

1. Stanley Osher has made influential contributions in a broad variety of fields in applied mathematics. These include high resolution shock capturing methods for hyperbolic equations, level set methods, PDE based methods in computer vision and image processing, and optimization. His numerical analysis contributions, including the Engquist-Osher scheme, TVD schemes, entropy conditions, ENO and WENO schemes and numerical schemes for Hamilton-Jacobi type equations have revolutionized the field. His level set contribu- tions include new level set calculus, novel numerical techniques, fluids and materials modeling, variational approaches, high codimension motion analysis, geometric optics, and the computation of discontinuous so- lutions to Hamilton-Jacobi equations; level set methods have been extremely influential in computer vision, image processing, and computer graphics. In addition, such new methods have motivated some of the most fundamental studies in the theory of PDEs in recent years, completing the picture of applied mathematics inspiring pure mathematics.

2. Stanley Osher has unique mentoring qualities: he has influenced the education of generations of outstanding applied mathematicians, and thanks to his entrepreneurship he has successfully brought his mathematics to industry.

Trained as an applied mathematician and an applied mathematician all his life, Osher continues to surprise the mathematical and numerical community with the invention of simple and clever schemes and formulas. His far-ranging inventions have changed our conception of physical, perceptual, and mathematical concepts, and have given us new tools to apprehend the world.

Chern Medalist:

Phillip Griffiths

[Phillip Griths is awarded the 2014 Chern Medal] for his groundbreaking and transformative development of transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties.

Phillip Griffiths’s ongoing work in algebraic geometry, differential geometry, and differential equations has stimulated a wide range of advances in mathematics over the past 50 years and continues to influence and inspire an enormous body of research activity today.

He has brought to bear both classical techniques and strikingly original ideas on a variety of problems in real and complex geometry and laid out a program of applications to period mappings and domains, algebraic cycles, Nevanlinna theory, Brill-Noether theory, and topology of K¨ahler manifolds.

A characteristic of Griffithss work is that, while it often has a specific problem in view, it has served in multiple instances to open up an entire area to research.

Early on, he made connections between deformation theory and Hodge theory through infinitesimal methods, which led to his discovery of what are now known as the Griffiths infinitesimal period relations. These methods provided the motivation for the Griffiths intermediate Jacobian, which solved the problem of showing algebraic equivalence and homological equivalence of algebraic cycles are distinct. His work with C.H. Clemens on the non-rationality of the cubic threefold became a model for many further applications of transcendental methods to the study of algebraic varieties.

His wide-ranging investigations brought many new techniques to bear on these problems and led to insights and progress in many other areas of geometry that, at first glance, seem far removed from complex geometry. His related investigations into overdetermined systems of differential equations led a revitalization of this subject in the 1980s in the form of exterior differential systems, and he applied this to deep problems in modern differential geometry: Rigidity of isometric embeddings in the overdetermined case and local existence of smooth solutions in the determined case in dimension 3, drawing on deep results in hyperbolic PDEs(in collaborations with Berger, Bryant and Yang), as well as geometric formulations of integrability in the calculus of variations and in the geometry of Lax pairs and treatises on the geometry of conservation laws and variational problems in elliptic, hyperbolic and parabolic PDEs and exterior differential systems.

All of these areas, and many others in algebraic geometry, including web geometry, integrable systems, and Riemann surfaces, are currently seeing important developments that were stimulated by his work.

His teaching career and research leadership has inspired an astounding number of mathematicians who have gone on to stellar careers, both in mathematics and other disciplines. He has been generous with his time, writing many classic expository papers and books, such as “Principles of Algebraic Geometry”, with Joseph Harris, that have inspired students of the subject since the 1960s.

Griffiths has also extensively supported mathematics at the level of research and education through service on and chairmanship of numerous national and international committees and boards committees and boards. In addition to his research career, he served 8 years as Duke’s Provost and 12 years as the Director of the Institute for Advanced Study, and he currently chairs the Science Initiative Group, which assists the development of mathematical training centers in the developing world.

His legacy of research and service to both the mathematics community and the wider scientific world continues to be an inspiration to mathematicians world-wide, enriching our subject and advancing the discipline in manifold ways.

Leelavati Prize Winner:

University of Buenos Aires, Argentina

[Adrian Paenza is awarded the Leelavati Prize] for his contributions have definitively changed the mind of a whole country about the way it perceives mathematics in daily life. He accomplished this through his books, his TV programs, and his unique gift of enthusiasm and passion in communicating the beauty and joy of mathematics.

Adrián Paenza has been the host of the long-running weekly TV program “Cient´ıficos Industria Argentina” (“Scientists Made in Argentina”), currently in its twelfth consecutive season in an open TV channel. Within a beautiful and attractive interface, each program consists of interviews with mathematicians and scientists of very different disciplines, and ends with a mathematical problem, the solution of which is given in the next program.

He has also been the host of the TV program “Alterados por Pi” (“Altered by Pi”), a weekly half-hour show exclusively dedicated to the popularization of mathematics; this show is recorded in front of a live audience in several public schools around the country.

Since 2005, he has written a weekly column about general science, but mainly about mathematics, on the back page of P´agina 12, one of Argentinas three national newspapers. His articles include historical notes, teasers and even proofs of theorems.

He has written eight books dedicated to the popularization of mathematics: five under the name “Matem´atica

. . . ¿est´as ah´ı?” (“Math . . . are you there?”), published by Siglo XXI Editores, which have sold over a million copies. The first of the series, published in September 2005, headed the lists of best sellers for a record of 73 consecutive weeks, and is now in its 22nd edition. The enormous impact and influence of these books has extended throughout Latin America and Spain; they have also been published in Portugal, Italy, the Czech Republic, and Germany; an upcoming edition has been recently translated also into Chinese.

失之毫厘，差之千里

1961年，作为天气预报员的Lorenz在利用计算机来做气象预测时，为了省事，就在第二次计算的时候，直接从第一次程序的中间开始运算。但是两次的预测结果产生了巨大的差异。Lorenz看到这个结果之后大为震惊，然后经过不断地测试，发觉在自己的模型当中，只要初始的数据不一样，就会产生不同的结果，而且结果大相径庭。在1979年的科学会议上，Lorenz简单的描述了“蝴蝶效应”:

$x^{'}(t) =\sigma(y-x)$

$y^{'}(t)=x(\rho-z)-y$

$z^{'}(t)=xy-\beta z$

分形几何学：复杂简单化

$z \mapsto z^{2}+c$

2. 刚性定理

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

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

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

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

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

(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。

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

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

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

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

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

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

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

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

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

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

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

1.一维动力系统中的双曲性

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

$|\lambda| \neq 1$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。

Axiom A： 假设 $f:[0,1]\rightarrow [0,1]$$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。

(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。

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

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

$\Rightarrow$ $\Lambda_{U}$ 是双曲集。

科普文：从人人网看网络科学（Network Science）的X个经典问题

http://blog.renren.com/share/270937572/16694767796?from=0101010202&ref=hotnewsfeed&sfet=102&fin=3&fid=24297752024&ff_id=270937572&platform=0&expose_time=1386225841

科普文：从人人网看网络科学（Network Science）的X个经典问题作者： 邓岳

长文，写了N个小时写完的。你肯定能看懂，所以希望你能看完，没看完就分享/点赞没有意义。有图有超链接，不建议用手机看。相关内容我想应该可以弄成一个小项目加到某门课中。

（1）最简单的指标（Common Neighbors，CN）

CN为两人共同好友的个数，直观感觉CN越大，此二人是好友的可能性越大。

CN(x,y)=|N(x)∩N(y)|        N(x)为节点x的所有邻居（计算原因也可以加上x自己）

（2）改进指标（Jaccard index）

Jaccard(x,y)=|N(x)∩N(y)| / |N(x)∪N(y)|

我刚注册人人，一个好友都没有，何谈共同好友？

这里还有问题大家可以考虑：推荐的都是目前和你没加好友的人，但整个人人里和你不是好友的有几千万人，总不能给这些人全都和你算一个共同好友数目，然后排序推荐。如何能圈定一个大致的范围？

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2、社团发现（Community Detection）

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3、中心性（Centrality）

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4、复杂网络的一些拓扑特性

（1）小世界（Small World）

（2）无标度（Scale-free）

（3）Giant Component

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1、首先删除网络中所有度为1的节点。删完以后检查，原来某些度大于1的节点会变成度为1的，就继续删，删完再检查，再删……直到没有度为1的节点为止。最终，认为刚才所有删掉的的节点属于第一层，即ks=1的节点（上图外围蓝色圈）；

2、现在网络中肯定没有度为1的节点了（都删掉了），那就开始删除度为2的节点（和上一步方法一致）。这次删掉的就是ks=2的节点（上图绿圈）；

3、依此类推，接着删度为3的节点，然后是度为4的节点……最终网络删干净了，网络中所有的点都被分配了一个ks值。

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6、网络的比对（alignment）、去匿名化（de-anonymization）

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7、动态网络（Dynamic Networks）/演化（Evolution）

（1）Growth：同一个社团，在相邻时刻出现了增长（如节点变多了）。和Growth相对的就是Contraction。

（2）Merging：t时刻的两个社团，在t+1时刻合并成了一个社团。和Merging相对的就是Splitting。

（3）Birth：t+1时刻新出现了一个社团（该社团在t时刻不存在）。和Birth相对的就是Death。

【未完待续】

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1、《网络科学导论》《链路预测》《网络科学》 、《Network Science》 入门且全面，正统的Network Science

2、《推荐系统实践》《推荐系统》 主流的Web应用里都有推荐系统，算是网络科学的主要应用方向

3、《网络、群体与市场》 也是入门书，结合经济学、社会学、计算与信息科学以及应用数学的有关概念与方法，考察网络行为原理及其效应。

4、《链接》 巴拉巴西早期经典著作

1、Social Network Analysis 2013.3开课时我全程跟下来并拿到了成绩。2013.10再次开课。强烈建议英文差不多的10级不考研同学10月跟一下这个。

2、网络、群体与市场 中文公开课，在Coursera也有。对软件方向的同学，建议重点看一下课程的第2、3、9、11、13~17章。

3、Social and Economic Networks: Models and Analysis，斯坦福，2014.1开课（来自文后留言）