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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.

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.

Ignore the zero-area boundary sets where a reciprocal or its digit is undefined, and put $a=\lfloor 1/x\rfloor$. On this digit branch the natural extension of the continued-fraction Gauss map has inverseThe images are the disjoint horizontal strips $1/(a+1)<y^{\prime }<1/a$, covering the square almost everywhere. Thus the branch calculation proves global invariance without an extra sum over overlapping branches.

Its Jacobian determinant has magnitude $J=[x^2(a+y{)}^2{]}^{-1}$. Sincethe transformed density satisfiesThe change of variables formula therefore proves invariance. Normalization follows fromProjecting onto the first coordinate commutes with the Gauss continued-fraction map, so its invariant marginal is the Gauss measure:One may also check it directly. The inverse branches $h_a(x)=(a+x{)}^{-1}$ givewhere the last sum telescopes. Values assigned at the exceptional boundary points do not affect the absolutely continuous invariant measure.

For a probability measure-preserving system, ergodicity means that every measurable set $A$ satisfying $T^{-1}A=A$ modulo null sets has measure zero or one. Equivalently, the only invariant functions are almost everywhere constant. The pointwise ergodic theorem states that, for $f\in L^1(\mu )$,where $\mathcal{I}$ is the invariant sigma-algebra; on a probability space the convergence also holds in $L^1$. For an ergodic transformation, the limit is the constant $\int fd\mu$.

For the Gauss continued-fraction map, the Gauss measure is the probability measureIts density integrates to one and is bounded above and below by positive constants, so it has the same null sets as Lebesgue measure. Apart from countably many endpoints, the inverse branches are $h_a(y)=1/(a+y)$, $a=1,2,\dots$. For a bounded measurable test function $F$, substitution on each branch givesThus $T$ is a measure-preserving transformation. Its definition at zero and the rational branch endpoints does not affect these identities.

For ergodicity, we give the full shrinking-cylinder argument. A digit cylinder with first $n$ partial quotients $a_1,\dots ,a_n$ is an interval $I=h((0,1))$, where $h=h_{a_1}\circ \cdots \circ h_{a_n}$ and $T^n\circ h$ is the identity. The continued fraction convergents giveHere $q_n\ge q_{n-1}>0$ after the initial step, and the recurrence $q_n=a_n{q}_{n-1}+q_{n-2}$ makes $q_n$ tend to infinity at least at the Fibonacci rate. Hence the nested cylinders of every irrational point shrink to that point. The bounded distortion of continued-fraction cylinders is uniform: the ratio of the largest to the smallest value of $|h^{\prime }|$ is at most four. Since $1/(1+h(y))$ ranges between $1/2$ and $1$, the densityhas maximum-to-minimum ratio at most eight. Consequently every measurable $E\subseteq (0,1)$ satisfiesThe last inequality uses $d\nu /dx\le 1/\log 2$.

Now suppose $T^{-1}A=A$ modulo null sets and $\nu (A)>0$. Choose an irrational density point $x\in A$, and let $I_n$ be its successive digit cylinders. The Lebesgue density theorem applies to these intervals even though they need not be centered: an interval of length $\ell$ containing $x$ lies in the centered interval of radius $\ell$, so its relative complement tends to zero. Comparability of the two densities then givesBut invariance gives $1_A(h(y))=1_A(y)$ almost everywhere for each inverse branch of $T^n$. Pulling the exceptional null sets through $h$ is legitimate since $h$ has a positive smooth Jacobian on this fixed branch. Thus the distortion estimate with $E=A^c$ givesLetting $n\to \infty$ forces $\nu (A^c)=0$. Therefore the Gauss continued-fraction map is ergodic for the Gauss measure.

For an irrational $x$, the successive continued fraction digits satisfy $a_j(x)=\lfloor 1/T^{j-1}x\rfloor$. The event that a digit equals two is the interval $(1/3,1/2]$, up to null endpoints. Apply the pointwise ergodic theorem to its indicator:This holds for almost every $x$ with respect to both the Gauss measure and Lebesgue measure, since they have the same null sets.

For $x>0$, $N(x)=\lfloor 1/x\rfloor$ and $T(x)=1/x-\lfloor 1/x\rfloor$. Iterating this Gauss continued-fraction map on an irrational number never reaches zero, and its continued fraction isThe branches $T:(1/(n+1),1/n)\to (0,1)$ have inverse $x_n(y)=1/(n+y)$, for $n\ge 1$. The change of variables formula for a probability density function gives, almost everywhere,The endpoints of these branches are a countable set of probability zero. Thus $T(X)$ and $X$ have the same distribution; this Gauss measure is an invariant measure for the Gauss continued-fraction map.