Myrberg's approximation theorem on Fuchsian groups.
Myrberg's approximation theorem on Fuchsian groups.
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Fuchsian 群的 Myrberg 近似定理。
DOI：
10.2969/jmsj/00430310
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发表时间：
1952
影响因子：
0.7
通讯作者：
M. Tsuji
中科院分区：
文献类型：
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作者：
M. Tsuji
$|z|<；1$，或者在$|z|=1$上具有有限数量的抛物线顶点。可以证明这等价于$D_{0}$的非欧几里得面积是有限的)。然后，Myrberg2)证明了如下逼近定理。定理。在$|z|=1$上存在度量为$2\pi$的集合$E$，它满足以下条件。设$L=L(\theta)$是通过$e^{i\theta}$的直径$|z|=1$，$i_{\triangleleft}$$(\nu=0，1，2，-)$是与$|z|=1$的任意正交圆。如果E$中的$e^{i\theta}\，则我们可以找到$\nu_{k}$，使得$L_{\nu_{k}}\right tarrow C(k\right tarrow\inty)$。我们将简单地用Hopf遍历定理证明这一定理。证据。我们表示$|z|=1$的正交圆，其在$|z|=1$上的终点为$e^{i\theta}，$$e^{i\varphi}$by$C(\theta，\varphi)$。现在，$(\theta，\varphi)$可以被认为是环面$\Omega$；$0\leqq\theta\leqq 2\pi，$$0\leqq\varphi\leqq 2_{pi}$上的一个点
$|z|<1$ , or has a finite number of parabolic vertices on $|z|=1$ . It can be proved that this is equivalent to that the non-euclidean area of $D_{0}$ is finitei). Then Myrberg2) proved the following approximation theorem. THEOREM. There exists a set $E$ of measure $ 2\pi$ on $|z|=1$ , which satisfies the following condition. Let $L=L(\theta)$ be a diameter of $|z|=1$ through $e^{i\theta}$ and $I_{\triangleleft}$ $(\nu=0,1,2, -)$ be its equivalents by G. Let $C$ be any orthogonal circle to $|z|=1$ . If $e^{i\theta}\in E$ , then we can find $\nu_{k}$ , such that $L_{\nu_{k}}\rightarrow C(k\rightarrow\infty)$ . We shall prove this theorem simply by means of Hopf’s ergodic theorem. PROOF. We denote an orthogonal circle to $|z|=1$ , whose end points on $|z|=1$ are $e^{i\theta},$ $e^{i\varphi}$ by $C(\theta, \varphi)$ . Now $(\theta, \varphi)$ can be considered as a point on a torus $\Omega$ ; $0\leqq\theta\leqq 2\pi,$ $0\leqq\varphi\leqq 2_{\pi}$ and the measure