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I have to

Prove that a loxodromic transformation has an attractor and a repeller as fixed points.

I have no idea how to start the proof, or what i need to do, formally. Basically, I only know the geometric picture of what an attractor and a repeller look like. Right now, it is beyond my scope of understanding, to figure out on my own what I have to do. So some help would be very appreciated, if there exists any.

$Def (loxodromic): \overline{T}$ lives in $PSL(2, \mathbb{C}).$ It is such that it fixes exactly 2 points in $\mathbb{\hat{C}}.$ $\overline{T}$ is conjugate in $PSL(2, \mathbb{C})$ to $\overline{S}(z) = \alpha z.$ If $|\alpha| \neq 1 $ and $\alpha \in \mathbb{R^+ }, \overline{T}$ is called loxodromic.

Trux
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    There's a lot of missing context in this question. What is the underlying space on which your transformation is acting? What is the definition of loxodromic? Without knowing that, I be very hesitant to answer this question. – Lee Mosher Feb 18 '18 at 14:29
  • $\overline{T}$ lives in $PSL(2, \mathbb{C}).$ It is such that it fixes exactly 2 points in $\mathbb{\hat{C}}.$ $\overline{T}$ is conjugate in $PSL(2, \mathbb{C})$ to $\overline{S}(z) = \alpha z.$ If $|\alpha| \neq 1 $ and $\alpha \in \mathbb{R^+ }, \overline{T}$ is called loxodromic. – Trux Feb 18 '18 at 19:23

1 Answers1

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Using the additional information given in your comment, here's what to do.

First, $\overline S$ has an attractor and repeller as fixed points. To see this, use the equation $\overline S^n(z) = a^n z$. If $\alpha > 1$ then $0$ and $\infty$ are the only fixed points, and for any other value of $z$ in the closed upper half plane one has $\lim_{n \to \infty} a^n = \infty$ and $\lim_{n \to -\infty} a^n = 0$, so $\infty$ is an attractor and $0$ is a repeller. Similarly, of $0<\alpha<1$ then $0$ is an attractor and $\infty$ is a repeller.

Second, if $\overline T = \overline R \overline S \overline R^{-1}$ and if $\alpha > 1$ then $\overline R(\infty)$ is the attractor for $T$ and $\overline R(0)$ is the repeller for $T$; and if $0 < \alpha < 1$ then $\overline R(0)$ is the attractor for $T$ and $\overline R(\infty)$ is the repeller for $T$.

Lee Mosher
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