On some symmetric multidimensional continued fraction algorithms

Abstract

We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear that it is of positive measure.

A Set of Sequences of Complexity $$2n+1$$ 2 n + 1

“…Many of its properties can be found in [11] and the density function of the invariant measure of f C was computed in [1].…”

“…Indeed, let w n = σ 0 (x) · · · σ n (x) (1). As c 1 and c 2 occur infinitely often, there exist infinitely many indices m such that σ m+1 (x) = c 1 and σ m+2 (x) = c 2 .…”

“…Proof. The directive sequence being primitive, all letters of A occur in w (1) . The result then follows from the fact that all morphisms τ in C ′ are either left proper (τ (A) ⊂ aA * for some letter a) or right proper (τ (A) ⊂ A * a for some letter a).…”

“…The next lemma describes how the extension set of a bispecial word determines the extension set of any of its bispecial extended images. (1) ) and if x, y ∈ A * are such that u = xτ 0 (v)y, then…”

“…For the inclusion ⊇, consider (a ′ , b ′ ) ∈ E(v) such that τ 0 (a ′ ) ∈ A * ax and τ 0 (b ′ )i ∈ ybA * . Let c ∈ A be such that a ′ vb ′ c is a factor of w (1) . Then τ 0 (a ′ vb ′ c) ∈ τ 0 (a ′ vb ′ )iA * ⊆ A * axτ 0 (v)ybA * is a factor of w and we have (a, b) ∈ E(u).…”