The Gauss map — continued fractions as orbits

Parent: orbit · Prereq: ergodic-theory-of-orbits

The most striking case of "an orbit is a classical object": iterating one map reads out the continued-fraction expansion, and Birkhoff turns that into digit statistics for almost every real number.

Gauss map

$G: (0,1) \to [0,1)$, $\quad G(x) = \dfrac{1}{x} - \left\lfloor \dfrac{1}{x} \right\rfloor$ (the fractional part of $1/x$). The digits of $x$ are $a_n(x) = \left\lfloor 1 / G^{n-1}(x) \right\rfloor$, and then

$$x = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}$$

— the orbit under $G$ shifts the continued fraction, exactly as the doubling map shifts binary digits.

The invariant measure (Gauss, 1812, in a letter — no proof left behind)

Lebesgue measure is not preserved. The right one:

Gauss measure

$\mu(A) = \dfrac{1}{\log 2} \displaystyle\int_A \dfrac{dx}{1+x}$ is $G$-invariant, and $G$ is ergodic with respect to it.

Verification of invariance on $A = (0, t)$: the preimage is $G^{-1}(0,t) = \bigcup_{k \ge 1} \left( \tfrac{1}{k+t}, \tfrac{1}{k} \right]$ (those $x$ whose $1/x$ has integer part $k$ and fractional part $< t$). Summing:

telescopes (write $\log(1+\tfrac1k) = \log(k+1) - \log k$ and $\log(1+\tfrac{1}{k+t}) = \log(k+1+t) - \log(k+t)$; partial sums collapse) to $\tfrac{\log(1+t)}{\log 2} = \mu\big((0,t)\big)$. ✓

What Birkhoff then buys — for almost every real number

Applying the ergodic theorem (ergodic-theory-of-orbits) to indicator and log functions of the digits:

• Gauss–Kuzmin digit frequencies: the digit $k$ appears with limiting frequency $$\mu\big(a_1 = k\big) = \log_2\!\left(1 + \frac{1}{k(k+2)}\right).$$ Count along: digit 1 → $\log_2 \tfrac{4}{3} \approx 41.50\%$, digit 2 → $\log_2 \tfrac{9}{8} \approx 16.99\%$, digit 3 → $\log_2 \tfrac{16}{15} \approx 9.31\%$. Small digits dominate, but every digit occurs with positive frequency — for almost every $x$.
• Khinchin's constant: the geometric mean of the digits converges, $\sqrt[n]{a_1 a_2 \cdots a_n} \to K_0 = \prod_{k\ge1} \left(1 + \tfrac{1}{k(k+2)}\right)^{\log_2 k} \approx 2.6854$ — the same constant for almost every real.
• Lévy's constant: the denominators $q_n$ of the convergents grow at a universal exponential rate, $\tfrac{1}{n}\log q_n \to \tfrac{\pi^2}{12 \log 2} \approx 1.1866$.

The fine print

"Almost every" excludes precisely the interesting arithmetic points: rationals (finite expansion — orbit hits $0$ and stops) and quadratic irrationals (eventually periodic digits — Lagrange's theorem; e.g. the golden ratio has $a_n \equiv 1$, geometric mean $1 \neq K_0$). Same shape of caveat as ergodic-theory-of-orbits and, at full strength, collatz-orbit-statistics: measure-theoretic knowledge, pointwise blindness.