The goals of this GeoGebra model are the following:
1) Input a hyperbolic isometry of the hyperbolic plane using the matrix representation. In other words input a real matrix with determinant 1 and trace more than 2.
2) Programmatically draw the axis.
3) Mark any point P on the hyperbolic plane (upper half plane)
4) Show the movement of the point P when powers of the isometry are applied sequentially
Here is a little video demonstration
Here is the Geogebra worksheet: