Gauss continued-fraction map ($T(x)=\{1/x\}$)

The map $T(x)=1/x-\lfloor 1/x\rfloor$ on $(0,1]$, with $T(0)=0$. Its digit $N(x)=\lfloor 1/x\rfloor$ gives the successive partial quotients $N(T^{j}x)$ of a continued fraction. This is distinct from the geometric Gauss map assigning surface normals.

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.

The inverse branch of the $n$th iterate of the Gauss continued-fraction map on a fixed digit cylinder has the displayed form, with continued fraction convergents $p_j/q_j$. Its derivative has absolute value $(q_n+q_{n-1}y{)}^{-2}$. Since $q_{n-1}\le q_n$, the maximum-to-minimum derivative ratio is at most four. The density of the Gauss measure changes this ratio by at most a further factor two. The cylinders shrink to irrational points, so this uniform distortion bound and the Lebesgue density theorem prove ergodicity of the map.

The invariant measure for the Gauss continued-fraction map with density $1/[(\log 2)(1+x)]$ on $[0,1]$. Inverse branches $x=1/(n+y)$ give transformed density $(\log 2{)}^{-1}{\sum }_{n\ge 1}[(n+y{)}^{-1}-(n+y+1{)}^{-1}]$, which telescopes to the same density.