On the dynamics of a complex continued fraction map which contains the Gauss map as its real number section

We consider the complex continued fraction map T defined by R. Kaneiwa, I. Shiokawa, and J. Tamura (1975) associated with the Eisenstein field $Q\left(\sqrt{-3}\right)$. A significant aspect of their continued fraction map is that the real number part of this map T is exactly the simple continued fraction map (Gauss map). In this paper we characterize the set of strictly periodic expansions of continued fraction expansions associated to this map in terms of quadratic extensions of $Q\left(\sqrt{-3}\right)$ in connection with the closure of $\{-\frac{q_n}{q_{n-1}}:n\ge 1\}$, where $q_n$, $n\ge 0$, is the denominator of the nth principal convergent of the continued fraction expansion. Moreover, we show that the closure of $\{-\frac{q_n}{q_{n-1}}:n\ge 1\}$ has positive Lebesgue measure on the complex plane $C$ though it has infinitely many holes. This gives us that the construction of the natural extension of T on a subset of $C^2\setminus \left\{\text{diagonal}\right\}$ is equivalent to the geodesics over the hyperbolic space $H^3$. Then the invariant measure for the natural extension map is given by the hyperbolic measure. Hence its projection to the complex plane is obviously the invariant measure for T.