# Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

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### Do smooth bijections always preserve Hausdorff dimension?

I am wondering if any smooth bijection $f \in C^\infty(\mathbb{C})$ on $\mathbb{C}$ preserves the Hausdorff dimension of any given subset $A \subset \mathbb{C}$? In particular, I am working on ...
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### Involution on triplets

Show that the map $\Phi$ on triplets $$(a,b,c) \overset{\Phi}{\mapsto} \left(\frac{ (b+c) a - 2 b c}{2 a -(b+c)}, \frac{(a+c)b-2 a c}{2 b-(a+c)}, \frac{(a+b)c- 2 a b}{2 c-(a+b)}\right)$$ is an ...
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### Domain and range of Mobius transformation

I read that the Mobius transformation $(z-i)/(z+i)$ is a biholomorphism from $\Re{z}>0$ to $B_1(0)$. How do I see this, and more generally, how do I determine the domains/ranges of general Mobius ...
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### Mobius transformations map $\overline{\mathbb{R}}$ to $\overline{\mathbb{R}}$ if and only if we can choose its coefficients to be real (proof check)

I am wanting to prove that a Mobius transformation $T(z)=\frac{az+b}{cz+d}$ with $ad-bc \neq 0$ maps $\overline{\mathbb{R}}$ to $\overline{\mathbb{R}}$ if and only if one can choose the coefficients a,...
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### Mobius Transform as functions of u(x,y) and v(x,y)

I am doing some plotting for my own interest of Mobius transforms, but my current system uses $u$ and $v$ axes where $u=u(x,y)$ and $v=v(x,y)$. I want to plot some Mobius transform f(z) as functions ...
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### Show that for every $z,w \in \mathbb{C}$ there exist $a,b,c,d \in \mathbb{R}$, s.t. $\frac{az+b}{cz+d}=w$ with $ad-bc=1$ and Im(z)>0 and Im(w)>0

Show that for every $z,w \in \mathcal{H}$ there exist $a,b,c,d \in \mathbb{R}$, s.t. $\frac{az+b}{cz+d}=w$ with $ad-bc=1$. This is just a small lemma in a bigger proof about a relation between ...
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### Möbius transformation composition

Suppose we have $F(z)=f(\phi(z))$ where $\phi$ is a mobius transformation which maps points of the unit circle to points of the unit circle. Suppose also that there is an interval of length $\pi$ such ...
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### How to prove that Linear-Fractional Transformations map circles/straight lines onto circles/straight lines?

There this theorem that states the Linear Fractional Transformations map circles and straight lines to circles and straight lines. How can you prove this? This is what the theorem states in my lecture ...
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