1
$\begingroup$

Again, is there a conformal self-map that interchanges two points in the upper half-plane? I'm beginning to think this isn't so. Such a map would be a FLT $\frac{az+b}{cz+d}, ad-bc=1$, with real coefficients. If I construct a map that sends some $w \mapsto t$ and $t \mapsto w$, I can't find values for the coefficients that work, though I may not have done this with sufficient generality.

I'm also thinking that given two points and considering a circle that they describe, the self-map would have to reverse the orientation of the circle which I don't think is possible.

  • 1
    Hm, you can certainly do it for some points. I'm a little rusty on my conformal mappings so I can't quite remember how much control you can get. But certainly you can map the half plane conformally to the disk and then do a rotation to swap any two points on opposite sides of the same line in the disk. Then pull back to the plane.2012-11-30
  • 0
    Oh, right, it may be better to think about this on the unit disk since it and the upper half-plane are isomorphic. I'll see what I can cook up, thanks.2012-11-30
  • 2
    In the unit disk there is always the Blaschke factor (maybe with a minus sign) that swaps $0$ and a point you choose.2012-11-30
  • 2
    And (also in the unit disk) $z \mapsto -z$ swaps everything in sight. :-) In the upper half plane, this corresponds to $f(z)=-1/z$.2012-11-30

1 Answers 1