Regarding existence of automorphism on the disc given two points

Question

Let $a,b,c,d$ be unique elements in the open unit disc $\mathbb{D}$ of the complex plane. Does there exist a unique automorphism $h$ on $\mathbb{D}$ such that $h(a)=c$ and $h(b)=d$?

I can give an automorphism that satisfies any one of the above condition, but not both.

This is with reference to an article that I have been reading. Where the second paragraph under Definition 1.20 demands such a function.

Answer 1

No, such an $h$ does not exist in general. Recall that every automorphism of $\mathbb{D}$ can be represented by a Mobius transformation of the form $$ h(z)=\frac{uz+v}{\overline{v}z+\overline{u}}$$ with $|u|^2-|v|^2=1$.

So taking $a=0$ for example, if $h(0)=c$ then $\frac{v}{\overline{u}}=c$, and so $h$ is determined up to multiplication by a complex number of modulus $1$. So any automorphism of $\mathbb{D}$ taking $0$ to $c$ has the form $$ h(z)=e^{i\theta}\frac{z+c}{\overline{c}z+1}$$ with $\theta\in\mathbb{R}$, and so $$ |h(b)|=\Big|\frac{b+c}{\overline{c}b+1}\Big|$$ for any such $h$. This means that given $b,c$, we can then choose $d$ with $d\neq |h(b)|$, making it impossible to have $h(b)=d$.

However, looking at the linked paper, I think they are asking for something weaker. The statement is: "there exists a unique automorphism $h$, such that $h(v)=0$ and $h(w)=a$, where $a\in[0,1)$". So I don't think $a$ is being specified exactly in this setup, rather it is simply required to be real.

Answer 2

The automorphism $h$ does not exist in general, but there is a rather beautiful necessary and sufficient condition for the existence of $h$.

It turns out that the automorphisms of $h$ are precisely the orientation preserving isometries of the unit disc $\mathbb D$ with respect to the Poincare metric, defined in real coordinates by the quadratic form $$ds^2 = \frac{4(dx^2 + dy^2)}{(1-x^2-y^2)^2} $$ This metric defines hyperbolic geometry, and it shares many of the properties of Euclidean geometry (except for the parallel postulate and its consequences). In particular this geometry is homogeneous in the same sense that Euclidean geometry is homogeneous: any two geodesic segments of the same length are congruent in the sense that there is an isometry taking one to the other; furthermore if you specify which endpoints go to which endpoints, and that the isometry is orientation preserving (which all automorphisms of $\mathbb D$ are), then it is unique.

Translating this into the language of your problem, $h$ exists if and only if the Poincare distance from $a$ to $b$ equals the Poincare distance from $c$ to $d$, in which case it is unique.