Continued fractions on the Heisenberg group

TL;DR

This paper extends the concept of continued fractions to the Heisenberg group, providing convergence estimates, analogs of classical formulas, and analyzing the dynamics of the associated Gauss map.

Contribution

It introduces a novel generalization of continued fractions to the Heisenberg group and explores their convergence and dynamical properties.

Findings

Explicit convergence rate estimates for the continued fractions on the Heisenberg group

Analogues of classical continued fraction formulas in this new setting

Comparison of the Gauss map dynamics with other expansion systems

Abstract

We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions. We then discuss dynamical properties of the associated Gauss map, comparing them with base-$b$ expansions on the Heisenberg group and continued fractions on the complex plane.