Ergodic Properties of Heisenberg Continued Fractions with Applications in Hyperbolic Geometry

We give an overview of how to construct continued fractions on the Heisenberg group $\mathbb{H}$, the projective and planar Siegel models of the group, and how to perform computations on the group using matrices. We discuss and work with some recent results of Vandehey on the classification of which elements of the Heisenberg group have eventually periodic continued fraction expansions.Then we examine and work with major theorems in ergodic theory to explore results concerning growth of denominators and digit frequency, relating the results to hyperbolic geometry.