ON DENJOY'S CANONICAL CONTINUED FRACTION EXPANSION

Abstract

where 0 ∈ Z is such that − 0 ∈ [0 1), and ∈ N for ≥ 1. As is well-known, the regular continued fraction (RCF) expansion of is finite if and only if ∈ Q. In this case there are two possible expansions, otherwise the expansion is unique. Apart from the RCF expansion there are very many other continued fraction expansions: the continued fraction expansion to the nearest integer, Nakada’s α-expansions, Bosma’s optimal expansion . . . in fact too many to mention (see [6] and [3] for some background information). One particular expansion, which attracted no attention whatsoever, and which is quite different from the continued fraction expansions mentioned above, is Denjoy’s canonical continued fraction expansion (see [2], or [1], p. 275–6 for the original paper by Denjoy). In [2], Denjoy stated that every real number has continued fraction expansions of the form