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?
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 $\alpha'=(\alpha_2',\alpha_3')$, $\beta'=(\beta_1',\beta_3')$, $\gamma'=(\gamma_1',\gamma_2')$.
Partial answer.
I imagine that you mean a map $\overline{G}$ which sends $([0;a_1,a_2,\ldots],[0;a_0,a_{-1},\ldots])$ on $([0;a_2,a_3,\ldots],[0;a_1,a_0,\ldots])$. If yes, we have a simple formula $$\overline{G}(x,y) = \big(\{1/x\},1/(\lfloor 1/x \rfloor + y)\big).$$ The probability measure on the set of all sequences $(a_n)_{n \in \mathbb{Z}}$ will be the stationary extension of the probability measure on the set of all sequences $(a_n)_{n \ge 1}$. I guess that there will be an invariant density measure on $[0,1[^2$, like the Gauss measure for $G$ on $[0,1[$, since the sequence $(A_n)_{n \ge 1}$ of partial quotients associated to a random real chosen according to the Gauss measure is very close to be an i.i.d. sequence, but I have no idea on whether it has a simple expression or not.
