On Denjoy's canonical continued fraction expansion (Q1866578)

In 1932 Denjoy showed that every real number \(x\) has so-called canonical continued fraction (CCF) expansions of the form \(x=[d_0;d_1,d_2,\ldots ]\), where \(d_0\in \mathbb Z\) such that \(x-d_0\geq 0\), where the digits \(d_i\) are \(0\) or \(1\) for \(i\geq 1\), and where \(d_i=0\) implies either that \(d_{i+1}=1\), or that \(d_j=0\) for all \(j>i\). Using certain operations on the digits of any positive number \(x\), it is shown that the regular continued fraction (RCF) expansion \(x=[a_0;a_1,a_2,\ldots ]\) can be converted to a unique CCF expansion: \[ x=[1;(1,0)^{a_0-1},1(1,0)^{a_1-1}, 1,\ldots ],\quad \text{if } a_0>0, \] and \[ x=[0;(1,0)^{a_1-1},1(1,0)^{a_2-1}, 1,\ldots ],\quad \text{if } a_0=0. \] From this the authors derive that positive rationals have two different finite CCF expansions of the above form, and that quadratic irrational have a CCF expansion which is (eventually) periodic. It is shown that the CCF convergents are all the RCF convergents \(p_n/q_n\)(with multiplicity \(a_{n+1})\) and mediant convergents \((ap_n+p_{n-1})/(aq_n+q_{n-1})\) (for \(0<a<a_{n+1}\)). A map \(T_d\) is given, similar to the Gauss-map, and it is shown that the measure \(\mu\), with density \(f\), given by \[ f(x)=\frac{1}{x}1_{(0,1]}(x)+\frac{1}{1+x}1_{[1,\infty )}(x),\quad x>0, \] is a \(\sigma\)-finite, infinite \(T_d\)-invariant measure. It is shown that the dynamical system \(( [0,\infty ), T_d, \mu )\) is ergodic. Although there are always more 1's than 0's in the CCF expansion, it follows from the relation with the RCF that for almost all \(x\) there are asymptotically always as many 0's as 1's.