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). 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
$\alpha'=(\alpha_2',\alpha_3')$,
$\beta'=(\beta_1',\beta_3')$,
$\gamma'=(\gamma_1',\gamma_2')$.