Ergodic Properties of Heisenberg Continued Fractions with Applications in Hyperbolic Geometry

TL;DR

This paper explores the ergodic properties of Heisenberg continued fractions, their construction, and applications in hyperbolic geometry, including recent classifications and growth behaviors.

Contribution

It provides a comprehensive overview of constructing Heisenberg continued fractions and connects ergodic theory results to hyperbolic geometric contexts.

Findings

Classification of elements with periodic continued fractions

Results on growth of denominators

Analysis of digit frequency in expansions

Abstract

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.