# Visualize Moebius Transformations

I want to visualize Moebius transformations, in particular the attractive and repulsive fixed points of a loxodromic Moebius transformation. I have had "some" success with ComplexListPlot, but when using Joined->True it all gets totally smudged up. And I'd rather see it with arrows, such as in StreamPlot or VectorPlot, ComplexVectorPlot, or ComplexStreamPlot.

Picking a specific example, 1/(2+w) is loxodromic and has an attractive and repulsive fixed point (there are up to two in a Moebius transformation, and exactly two for loxodromic). How can I visualize that for a grid of complex input?

Something like

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Elliptic_transformations

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Hyperbolic_transformations

I know the rules of this site require the o/p to show what he tried. All I can say is that ComplexListPlot in principle "works", but leaves a total mess. I have also tried computing derivatives, because StreamPlot and VectorPlot work on derivatives, but I couldn't find a way to generalize this enough, and the Moebius transformations have nothing to do with derivatives as such.

If I can't get arrows, that would be fine. But I definitely want lines that show the attractive and repulsive nature of loxodromic fixed points.

• All I can say is that ComplexListPlot in principle "works", but leaves a total mess. - it could still help to show the corresponding code though. It would provide a starting point at least, and it would explain what you consider a mess so others don't reproduce it. Sep 4, 2022 at 21:14

ComplexPlot[(z - I/2)/(1 + I/2*z), {z, -3 - 3 I, 3 + 3 I},
Mesh -> {Range[-5, 5], Range[-5, 5]},  MeshFunctions -> {Re[#2] &, Im[#2] &},
MeshStyle -> {White, Black}]


Addition. Here is a simple visualization of an attractive fixed point of a Moebius transform in the complex plane.

f[z_] := (z - I/2)/(1 + I/2*z)
ComplexListPlot[Table[Nest[f, -3, n], {n, 1, 20}]]


We see the iterated points of -3 tend to -I. In order to produce the pictures in the "Iterating a transformation" section from the second link the TransformedRegion command may be applied.

• Hmmmmm ... is this supposed to tell me visually that the attractive fixed-point of your function is at -I? I don't see arrows/curves that show the fixed points of Moebius transformations in this. Is it possible you didn't understand my question? Please convince me that this addresses the fixed-points of a Moebius transformation, because I hate to downvote. Sep 5, 2022 at 14:38