Natural extension of the continued-fraction Gauss map ($F(x,y)=(\{1/x\},(\lfloor 1/x\rfloor +y{)}^{-1})$)

On each branch with digit $a=\lfloor 1/x\rfloor$, the displayed extension of the Gauss continued-fraction map is invertible almost everywhere: $x=(a+x^{\prime }{)}^{-1}$ and $y=(y^{\prime }{)}^{-1}-a$, with $a=\lfloor 1/y^{\prime }\rfloor$. Its invariant joint probability density function is $[(\log 2)(1+xy{)}^2{]}^{-1}$. The branch Jacobian determinant is $[x^2(a+y{)}^2{]}^{-1}$, which cancels the transformation of $1+xy$. Integrating out $y$ gives the Gauss measure density.