| Abstract: |
| We consider the nearest integer complex continued fraction map associated to
the Euclidean fields $\mathbb Q(\sqrt{-d})$, $d = 1, 2, 3, 7, 11$.
For each map, we see that there is an absolutely continuous ergodic invariant probability measure.
We will explain how to construct the natural extension of each map
on a subset of $\mathbb C \times \mathbb C$. Then the invariant measure for
this extension is derived from the hyperbolic measure on $\mathbb H^{3}$
and the density function of the absolutely continuous invariant measure is given as its marginal. |
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