Monatshefte für Mathematik

, Volume 188, Issue 1, pp 37–86 | Cite as

On the construction of the natural extension of the Hurwitz complex continued fraction map

  • Hiromi Ei
  • Shunji Ito
  • Hitoshi NakadaEmail author
  • Rie Natsui


We consider the Hurwitz complex continued fraction map associated to the Gaussian field \({\mathbb {Q}}(i)\). We characterize the density function of the absolutely continuous invariant measure for the map associated to the Hurwitz continued fractions. For this reason, we construct a representation of its natural extension map (in the sense of an ergodic measure preserving map) on a subset of \({\mathbb {C}} \times {\mathbb {C}}\). This subset is constructed by the closure of pairs of the n-th iteration of a complex number by the Hurwitz complex continued fraction map and \(-\frac{Q_{n}}{Q_{n-1}}\), where \(Q_{n}\) is the denominator of the n-th convergent of the Hurwitz continued fractions. The absolutely continuous invariant measure for the natural extension map is induced from the invariant measure for Möbius transformations on the set of geodesics over three dimension upper-half space. Then the absolutely continues invariant measure for the Hurwitz continued fraction map is given by its marginal measure.


Complex continued fractions Invariant measure Natural extension Tiling 

Mathematics Subject Classification

Primary 11K50 37A45 37D40 Secondary 05B45 52C20 



The authors would like to thank referees for their helpful comments. They also would like to thank Masahiro Mizutani, who carried out the computer simulation of \(\{ \frac{Q_{n-1}}{Q_{n}}\}\) (Fig. 10) based on the second author’s idea in the early ’80th. The first author was partially supported by JSPS Grants No. 23540141. The third author was partially supported by JSPS Grants No. 16K13766 and JSPS Core-to-core program, “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”. The fourth author was partially supported by JSPS Grants No. 15K17559.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Science and TechnologyHirosaki UniversityBunkyo-cho, HirosakiJapan
  2. 2.Higashimurayama-shiTokyoJapan
  3. 3.Department of MathematicsKeio UniversityKohoku-ku, YokohamaJapan
  4. 4.Department of Mathematics and PhysicsJapan Women’s UniversityBunkyo-kuJapan

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