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

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

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

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

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

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

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