Arithmetic and ergodic properties of 'flipped' continued fraction algorithms
where a0 ∈ Z is such that x − a0 ∈ [0, 1), i.e. a0 = bxc, and an ∈ N for n ≥ 1. The expansion (1), abbreviated by x = [0; a1, a2, . . . , an, . . . ], is infinite if and only if x ∈ R \ Q; in case x ∈ Q—essentially due to Euclid’s algorithm—x has two finite expansions. The partial quotients an are given by (2) an = an(x) = b1/Tn−1(x)c if Tn−1(x) 6= 0, where T : [0, 1) → [0, 1) is the continued fraction (or Gauss) map, defined by