Classification of Mobius transformations
I'm studying for an exam, and a point I'm unclear on is how to classify Mobius transformations into the different subcategories for lambda > 0.
I know the transformations are classified by the number of fixed points and the value of the constant lambda when the transformation are written in normal form. In my notes, I have that the transformation is
- elliptic when there are 2 fixed points and lambda = e ^ (i * theta0
- hyperbolic when there are 2 fixed points and lambda > 0
- loxodromic when there are 2 fixed points and lambda = r * e ^ (i * theta) when r is not 1
- parabolic when there is 1 fixed point
So my question is: If I have found two fixed points and a value of lambda greater than 0, how can I tell if it is hyperbolic or loxodromic just based on normal form? If lambda > 0 canít that technically always be written as r * e ^ (i * theta) for some r?