Calculations of the invariant measure for Hurwitz Continued Fractions

TL;DR

This paper develops a computational method to analyze the invariant measure density of Hurwitz complex continued fractions, providing Taylor coefficients, conjectures, and digit admissibility results, advancing understanding of this mathematical object.

Contribution

It introduces a new computational approach for approximating the invariant measure density and explores its properties, including digit admissibility and conjectures about its behavior.

Findings

Computed Taylor coefficients of the density at specific points

Proposed new conjectures on the density's behavior

Detailed characterization of admissible digit strings

Abstract

We study the density of the invariant measure of the Hurwitz complex continued fraction from a computational perspective. It is known that this density is piece-wise real-analytic and so we provide a method for calculating the Taylor coefficients around certain points and also the results of our calculations. While our method does not find a simple "closed form" for the density of the invariant measure (if one even exists), our work leads us to some new conjectures about the behavior of the density at certain points. In addition to this, we detail all admissible strings of digits in the Hurwitz expansion. This may be of independent interest.