DYNAMICAL GENERALIZATIONS OF THE LAGRANGE SPECTRUM

Sébastien Ferenczi (1)

1 IML - Institut de mathématiques de Luminy

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne n , where e n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of 1 nen. The Lagrange spectrum is the set of finite values of L(α) for all irrational numbers α, where L(α) is the largest constant c such that |α − p q | ≤ 1 cq 2 for infinitely many integers p and q (a variant is known as the Markov spectrum, see Section 1.3 below). It was recently remarked that this arithmetic definition can be replaced by a dynamical definition involving the irrational rotations of angle α, through their natural coding by the partition {[0, 1 − α[, [1 − α, 1[}. Namely, as we prove in Theorem 2.4 below which was never written before, L(α) is also the upper limit of the inverse of the so-called Boshernitzan's ne n , where e n is the smallest (Lebesgue) measure of the nonempty cylinders of length n. Thus, for any symbolic dynamical system, it is interesting to compute two new invariants of topological conjugacy, lim sup n→+∞ 1 nen

Domaines: Mathématiques [math]

Sébastien Ferenczi. DYNAMICAL GENERALIZATIONS OF THE LAGRANGE SPECTRUM. Journal d'analyse mathématique, 2012. ⟨hal-01265514⟩