# 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。

# 245A: Problem solving strategies

This is going to be a somewhat experimental post. In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a dozen or so basic tricks and techniques that are used over and over again. But I had not thought to actually try to make these tricks explicit, so I am going to try to compile here a list of some of these techniques here. But this list is going to be far from exhaustive; perhaps if other recent students of real analysis would like to share their own methods, then I encourage you to do so in the comments (even – or especially – if the techniques are somewhat vague and general in nature).