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

Abstract

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 Mobius 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.