The n-dimensional Stern–Brocot tree

This paper generalizes the Stern–Brocot tree to a tree that consists of all sequences of [Formula: see text] coprime positive integers. As for [Formula: see text] each sequence [Formula: see text] is the sum of a specific set of other coprime sequences, its Stern–Brocot set [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in [Formula: see text] and a multiresolution partition of [Formula: see text]. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern–Brocot set forms the matrix columns. This induces a finitely generated submonoid [Formula: see text] of [Formula: see text], and a unimodular multidimensional continued fraction algorithm, also generalizing [Formula: see text]. It turns out that the [Formula: see text]-dimensional subtree starting with a sequence [Formula: see text] is isomorphic to the entire [Formula: see text]-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation.

On some symmetric multidimensional continued fraction algorithms

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 the 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 whether it is of positive measure.

On the periodic writing of cubic irrationals and a generalization of Rédei functions

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

A combinatorial approach to products of Pisot substitutions

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

A generalized family of multidimensional continued fractions: TRIP Maps

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations

The two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, such as Jacobi–Perron, Poincaré, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.