Isometric Circle

Description

Given a Möbiustransformation on the upper half plane there is always a circle whose euclidian radius is preserved. This circle plays an important role when understanding the Modular group.

Isometric circles are also important for the Ford fundamental region, as they give a unique way of defining a Dirichlet region.

Definition

Let $T(z)=\frac{az+b}{cz+d}\in PSL(2,\mathbb{R})$ be an element of the modular group where $c\ne 0$. The circle $I(T)=\{z\in \mathbb{C}\mid |cz+d|=1\}$ is called the isometric circle. This circle is centered on the real number $\frac{d}{c}$.

Idea: To find the points at which no scaling happens, we calculate the derivative $|T^{\prime }(z)|=|cz+d{|}^{-2}$. This is equal to $1$ exactly if $z$ lies in a circle $|z+\frac{d}{c}|=\frac{1}{|c|}$.

This definition only works for the hyperbolic half plane but can easily be adapted to the Poincaré Scheibe-Model.

Note: Isometric circles are weird since they are not the same under iteration of the map. Moving a circle multiple times will actually change its size after the second time.

Properties

Length-increases and decreases

The transformation $T$ increases Euclidean lenghts and areas inside the isometric circle and decreases them outside.

This is kind of hard to imagine. How does the circle move exactly?

Describe the Limit set

Let $\Gamma$ be a Fuchsian group. Consider the set of all isometric circle centres. ${\Lambda }_0(\Gamma )$ is the set of (non-trivial) limit points in this set. This is exactly the Limit set.

Examples

Example