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 $T$.