Let $G:(0,1)\to(0,1)$ be the Gauss map, i.e., $G(x)=\left\{\frac1{x}\right\}$, which is known to act as the shift on the space of continued fraction expansions.


**Question.** Is there an explicit expression for the natural extension of $G$ in $\mathbb R^2$? What about its density?



Natural extension of the Gauss map
As it was already mentioned by Christophe Leuridan for classical continued fraction
we have extended Gauss measure
$$d\overline{\nu}(x,y)=\frac{1}{\log2}\cdot\frac{dx\,dy}{(1+xy)^2}=\frac{1}{\log2}\cdot\begin{vmatrix} 1 & \mp x\\
\pm y & 1
\end{vmatrix}^{-2} dx\,dy,\qquad (x,y)\in[0,1]^2,$$
which is invariant under the map
$$\overline{T}:(x,y)\to\begin{cases}(\{1/x\},([1/x]+y)^{-1},&\text{ if }x\ne 0;\\
(x,y),&\text{ if }x= 0.
\end{cases}$$
Details can be found in the book
*M. Iosifescu and C. Kraaikamp, Metrical theory of continued fractions, 2002.*


From geometrical point of view this extended Gauss measure describes the distribution of reduced Voronoi bases $(b\_1,b\_2)$ with $b\_1=(1,\mp x)$, $b\_2=(\pm y,1)$ on the boundary of the unit square.


In dimension $3$ something similar is knowт for Voronoi-Minkowski continued fractions(see [Three-dimensional continued fractions and Kloosterman sums](https://iopscience.iop.org/article/10.1070/RM2015v070n03ABEH004953/meta)). The measure has the form
$$\mathrm{const} \int\_{\Pi}\frac{d\alpha'd\beta'd\gamma'}{(\det X)^3},$$
where $X=\begin{pmatrix}
 1 & \alpha\_2' & \pm\alpha\_3'\\
-\beta\_1' & 1 & \beta\_3'\\
 \gamma\_1'&-\gamma\_2'&1\\
\end{pmatrix}$ are matricies of reduced Minkowski bases and
$\alpha'=(\alpha\_2',\alpha\_3')$,
$\beta'=(\beta\_1',\beta\_3')$,
$\gamma'=(\gamma\_1',\gamma\_2')$.


