= Natural extension of the continued-fraction Gauss map
{title2=$F(x,y)=(\{1/x\},(\lfloor1/x\rfloor+y)^{-1})$}
On each branch with digit $a=\lfloor1/x\rfloor$, the displayed extension of the <Gauss continued-fraction map> is invertible almost everywhere: $x=(a+x')^{-1}$ and $y=(y')^{-1}-a$, with $a=\lfloor1/y'\rfloor$. Its invariant joint <probability density function> is $[(\log2)(1+xy)^2]^{-1}$. 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.
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