An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. Indeed, with a word is naturally associated , whenever it exists, the vector of frequencies of its letters. A… (More)
We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincaré maps of measured foliations a suitable definition of return words which yields that the set of first return words to a given word is a symmetric basis of the free group on the underlying alphabet A. The set of first return… (More)
We study balancedness properties of words given by the Arnoux-Rauzy and Brun multi-dimensional continued fraction algorithms. We show that almost all Brun words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely balanced; in particular, bound-edness of the strong partial quotients implies balancedness. On the other hand, we provide… (More)
We introduce specular sets. These are subsets of groups which form a natural generalization of free groups. These sets of words are an abstract generalization of the natural codings of interval exchanges and of linear involutions. We consider two important families of sets contained in specular sets: sets of return words and maximal bifix codes. For both… (More)
Nous construisons des solutions bornées auxéquations intégrales λx 0 f (t) dt = f (x) − f (0) o` u λ ≥ 2 est un entier. Cette construction s'appuie sur une méthode originale de limite de sommes de Birkhoff itérées.
We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface, and show that the… (More)