Sequences and dynamical systems associated with canonical approximation by rationals (Q2895958)

For irrational \(x \in (0,1)\), let \(p_n/q_n\) be the \(n\)th approximate of the regular continued fraction expansion of \(x\). The quality of diophantine approximation is given by \(\theta_n = q_n| q_n x - p_n|\). For each \(\alpha \in (0,1]\), the author studies certain dynamical systems related to the subsequence \(p_{n_k}/q_{n_k}\) defined by \(\theta_{n_k}< \alpha\). The author combines a hyperbolic geometric viewpoint with the ergodic theory approach of Jager and co-authors (in particular to prove the so-called Doeblin-Lenstra conjecture, see [\textit{W. Bosma, H. Jager} and \textit{F. Wiedijk}, Indag. Math. 45, 281--299 (1983; Zbl 0519.10043)]). The well written work ends with some tantalizing conjectures to the effect that the ``complexity'' of the dynamical systems studied here might depend upon \(\alpha\) in a manner reminiscent of the Markoff spectrum.