On continued fraction maps associated with a submodule of $\mathfrak o(\sqrt{-3})$

We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = \eta \cdot\mathfrak o(\sqrt{-3})$, $\eta = \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map associated with $\mathfrak o(\sqrt{-1}) \cdot (1 + i)$ defined by J.~Hurwitz in the case of the Gaussian field. Together with $T$, we show that all complex numbers $z$ can be expanded as $\mathcal J$-coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent $q_{n}$ and the existence of the absolutely continuous ergodic invariant probability measure for $T$.