Find Mobius Transformations (M→N)

1. Apr 22, 2012

tazzzdo

1. The problem statement, all variables and given/known data

H is the upper-half plane model of the hyperbolic space

Find all Mobius transformations that send M to N.

2. Relevant equations

a) M = {0, 1, ∞}, N = {∞, 0, 1}
b) M = {0, 1, ∞}, N = {0, ∞, 2}
c) M = {i, -i, 3i}, N = {∞, i + 1, 6}

3. The attempt at a solution

Using the transformation:

Could I develop two equations m(z) and n(z) that map M and N to {0, 1, ∞}, respectively? Then just find n-1$\circ$m? Wouldn't that result in M→N?

If that's true, I'm just confused as to what points to use for z1, z2, z3.

2. Apr 22, 2012

Dick

That works. Your transformation f1(z) takes z1->0, z2->1 and z3->∞. Which case of finding a specific transformation m(z) or n(z) are you confused about? Are you supposed to map the sets in the given order or in any order?

Last edited: Apr 22, 2012
3. Apr 22, 2012

tazzzdo

Well if I plug those z's in, I get z(-∞)/(z-∞) for m(z). I seem to hit a wall there. Same with n(z). I don't know how to fix those equations into a form that I can use for the composition.

4. Apr 22, 2012

Dick

Which part are you working on? I suspect it's trying to map {0,1,∞} to {0,1,∞}. If z is finite then (-∞)/(z-∞) should cancel to 1. That would mean m(z)=z should work. It does, doesn't it?

5. Apr 22, 2012

tazzzdo

Yeah I see what you're saying. Then does that mean I just switch the z's around in (a) for n(z)? But use the same transformation?

6. Apr 23, 2012

tazzzdo

I cant find a transformation n such that n(1) = infinity

7. Apr 23, 2012

Dick

Why not? Find it the same way you found m(z). What do you get when you put the z1, z2, z3 in?