Flots chaînés

Isabelle Abou (1) , Pierre Liardet (2)

1 Lycée français J. Supervielle
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We investigate the dynamical structures of flows arising from arithmetic sequences, called chain sequences, computed from digital expansion of integers in a given base. A classical example in base 2 is the sequence n → αs_w(n) mod 1 where α is irrational and s_w(n) is the number of occurrences of the binary word w (not written with the only digit 0) in the binary expansion of n. We distinguish contractive and non contractive chained sequences. In the above example, the sequence is contractive if w is of length ≥ 2 and non contractive but completely 2-additive mod 1 if w = 1. The flows associated to such sequences are minimal and uniquely ergodic. In the contractive case the flow is metrically conjugated to a skew product. The non contractive case is more intricated but analogous. In both cases the spectral type of the underlying dynamical system is classified and applications to uniform distribution are given.

Mots clés: digital sequence; dynamical system

Domaines: Théorie des nombres [math.NT]

Isabelle Abou, Pierre Liardet. Flots chaînés. Proceedings of the Sixth Congress of Romanian Mathematicians, 2009, Vol. 1, pp.401-432. ⟨hal-00871066⟩