Calculations of the Invariant Measure for Hurwitz Continued Fractions

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.

Keywords:

• calculation of invariant measure
• Hurwitz complex continued fraction
• admissible sequences

1991 Mathematics Subject Classification:

• 37M25 (11Y65
• 11K50)

Acknowledgments

The authors thank the referee for their suggestions in developing the section on entropy.

Notes

1 Brothers Adolf and Julius Hurwitz each have their own complex continued fraction expansion [Oswald and Steuding, 2014]. We will be considering the expansion investigated by Adolf Hurwitz.

2 Although we use somewhat similar notation to the Ei, et al., paper, our definitions are distinct and should not be mistaken for one another.

Additional information

Funding

The first author is partially supported by the National Science Foundation under agreements No. DMS1406190.