Description
An Isometrie is called axial if it acts like a translation along a Geodesic.
Definition
Let be a Metrischer Raum. An Isometrie is called axial if there is a bi-infinite geodesig on which acts as a nontrivial translation.
Note, that an element might have many axes, which identify as the same Quasi-geodesic and therefore will be only a finite distance apart. One theorem, similar but stronger to the Morse lemma states that all axis are contained within a -neighbourhood of another axis.
Properties
Parallel axes in hyperbolic space
Let be a -Hyperbolic metric space. Then any two axes of are at most apart.
Examples
Hyperbolic möbius transformation
The Hyperbolische Möbiustransformation is a good example of an axial isometry.