## 数学代写|复分析作业代写Complex function代考|Fixed Points

As another step towards establishing (3.25), let us show that if a Möbius transformation exists mapping three given points to three other given points, then it is unique. To this end, we now introduce the extremely important concept of the fixed points of a Möbius transformation. Quite generally, $p$ is called a fixed point of a mapping $f$ if $f(p)=p$, in which case one may also say that $p$ is “mapped to itself”, or that it “remains fixed”. Note that under the identity mapping, $z \mapsto \mathcal{E}(z)=z$, every point is a fixed point.

By definition, then, the fixed points of a general Möbius transformation $M(z)$ are the solutions of
$$z=M(z)=\frac{a z+b}{c z+d}$$
Since this is merely a quadratic in disguise, we deduce that
With the exception of the identity mapping, a Möbius transformation has at most two fixed points.
From the above result it follows that if a Möbius transformation is known to have more than two fixed points, then it must be the identity. This enables us to establish the uniqueness part of (3.25). Suppose that $M$ and $N$ are two Möbius transformations that both map the three given points (say $q, r, s$ ) to the three given image points. Since $\left(N^{-1} \circ M\right)$ is a Möbius transformation that has $q, r$, and $s$ as fixed points, we deduce that it must be the identity mapping, and so $N=M$. Done.
We now describe the fixed points explicitly. If $M(z)$ is normalized, then the two fixed points $\xi_{+}, \xi_{-}$are given by [exercise]
$$\xi_{ \pm}=\frac{(a-d) \pm \sqrt{(a+d)^2-4}}{2 c}$$
In the exceptional case where $(a+d)= \pm 2$, the two fixed points $\xi_{ \pm}$coalesce into the single fixed point $\xi=(a-d) / 2 c$. In this case the Möbius transformation is called parabolic.

Let us now briefly outline how the fixed point can be used to classify the Möbius transformations into just four achetypes. The full mathematical details are worked out later, in Section 3.7. Recall our earlier observation (3.2), that, miraculously, the Möbius transformations exactly correspond to the symmetries (called Lorentz transformations) of Minkowski and Einstein’s spacetime. Thus this classification of the Möbius transformations is important for relativity theory, too! We cannot explore this in detail here, but see Needham $(2021, \$ 6.4)$for the technical details of the spacetime interpretation via Lorentz transformations Provided$c \neq 0$then the fixed points both lie in the finite plane; we now discuss the fact that if$c=0$then at least one fixed point is at infinity. If$c=0$then the Möbius transformation takes the form$M(z)=A z+B$, which represents, as we have mentioned, the most general “direct” (i.e., conformal) similarity transformation of the plane. If we write$A=\rho e^{i \alpha}$then this may be viewed as the composition of an through these fixed points (which are orthogonal to the invariant circles) are permuted among themselves. This pure rotation is the simplest, archetypal example of a so-called elliptic Möbius transformation. With$\rho>1$, figure [3.26b] illustrates the induced transformation on$\Sigma$corresponding to the origin-centred expansion of$\mathbb{C}, z \mapsto \rho z$. If$\rho<1$then we have a contraction of$\mathbb{C}$, and points on$\Sigma$move due South instead of due North. Again it is clear that the fixed points are 0 and$\infty$, but the roles of the two families of curves in [3.26a] are now reversed: the invariant curves are the great circles through the fixed points at the poles, and the orthogonal horizontal circles are permuted among themselves. This pure expansion is the simplest, archetypal example of a so-called hyperbolic Möbius transformation. # 复分析代考 ## 数学代写|复分析作业代写Complex function代考|Fixed Points 作为建立 (3.25) 的另一个步骤，让我们证明如果存在将三个给定点映射到另外三个给定 点的莫比乌斯变换，那么它是唯一的。为此，我们现在介绍莫比乌斯变换的不动点伩个 极其重要的概念。一般而言，$p$称为映射的不动点$f$如果$f(p)=p$，在这种情况下，也 可以说$p$被“映射到自身”，或者它”保持固定”。请注意，在身份映射下，$z \mapsto \mathcal{E}(z)=z$， 每个点都是不动点。 那么，根据定义，一般莫比乌斯变换的不动点$M(z)$是解决方案 $$z=M(z)=\frac{a z+b}{c z+d}$$ 由于这只是变相的二次方程，我们推断 除了恒等映射外，莫比乌斯变换至多有两个不动点。 从上面的结果可以看出，如果已知莫比乌斯变换有两个以上的不动点，那么它一定是恒 等式。这使我们能够建立 (3.25) 的唯一性部分。假设$M$和$N$是两个 Möbius 变换，它们] 都映射三个给定点（比如$q, r, s)$到三个给定的图像点。自从$\left(N^{-1} \circ M\right)$是一个莫比乌 斯变换，它有$q, r$，和$s$作为固定点，我们推断它必须是恒等映射，所以$N=M$. 完 毕。 我们现在明确地描述不动点。如果$M(z)$归一化，那么两个不动点$\xi_{+}, \xi_{-}$由[练习]给出 $$\xi_{ \pm}=\frac{(a-d) \pm \sqrt{(a+d)^2-4}}{2 c}$$ 在特殊情况下$(a+d)= \pm 2$, 两个不动点$\xi_{ \pm}$合并成一个固定点$\xi=(a-d) / 2 c$. 在这种 情况下，莫比乌斯变换称为抛物线变换。 ## 数学代写|复分析作业代写Complex function代考|Fixed Points at Infinity 现在让我们简要概述如何使用不动点将莫比乌斯变换分类为四种原型。完整的数学细节稍后在第 3.7 节中计算出来。回想一下我们之前的观察（3.2），莫比乌斯变换奇迹般地恰好对应于闵可夫斯基和爱因斯坦时空的对称性（称为洛伦兹变换）。因此，这种莫比乌斯变换的分类对于相对论也很重要！我们不能在这里详细探讨这一点，但请参阅 Needham(2021,$6.4)通过洛伦兹变换的时空解释的技术细节

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