Gauss map

Example

The Gauss map is $$G \colon [0, 1) \to [0, 1), \qquad G(x) = \frac{1}{x} - \floor{\frac{1}{x}} \text{ for } x \ne 0, \qquad G(0) = 0.$$ If $x = [0; a_1, a_2, a_3, \dots]$ is the continued fraction expansion of an irrational $x \in (0, 1)$, then $a_1 = \floor{1/x}$ and $G(x) = [0; a_2, a_3, \dots]$. So $G$ shifts the continued fraction digits, as the doubling map shifts binary digits.

The Gauss measure is the probability measure $$\mu(A) = \frac{1}{\log 2} \int_A \frac{dx}{1 + x}.$$ It is $G$-invariant; check on $A = [0, t)$ with $0 < t \le 1$. The preimage $G^{-1}[0, t)$ is, up to the point $0$, the disjoint union over $n \ge 1$ of the intervals $\big(\frac{1}{n + t}, \frac{1}{n}\big]$. The measure of the $n$-th interval is $\frac{1}{\log 2}\big(\log\frac{n+1}{n} - \log\frac{n+t+1}{n+t}\big)$. The partial sum up to $N$ telescopes to $\frac{1}{\log 2}\big(\log(N+1) - \log(N+1+t) + \log(1 + t)\big)$, which tends to $\frac{\log(1+t)}{\log 2} = \mu([0, t))$.

The map $G$ is ergodic for $\mu$, and $\mu$ is equivalent to Lebesgue measure. Hence for Lebesgue almost every $x$, the digit $k$ appears among the $a_n$ with frequency $\frac{1}{\log 2}\log\frac{(k+1)^2}{k(k+2)}$, the Gauss–Kuzmin distribution. The transfer operator of $G$ for Lebesgue measure is the Gauss–Kuzmin–Wirsing operator, $(\mathcal{L}\varphi)(x) = \sum_{n \ge 1} \frac{1}{(x+n)^2} \varphi\big(\frac{1}{x+n}\big)$, and the density $\frac{1}{(1+x)\log 2}$ is its fixed point.