Geometry, dynamics, and arithmetic of $S$-adic shifts

Valérie Berthé (1) , Wolfgang Steiner (1) , Jörg M. Thuswaldner (2)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 MUL - Montanuniversität Leoben

This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions in an adic way. Pure discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux-Rauzy as well as Brun substitutions (related to the respective continued fraction algorithms). It is shown that almost all of these shifts have pure discrete spectrum, which proves a conjecture of Arnoux and Rauzy going back to the early nineties in a metric sense. We also prove that each linearly recurrent Arnoux-Rauzy shift with recurrent directive sequence has pure discrete spectrum. Using $S$-adic words related to Brun's continued fraction algorithm, we exhibit bounded remainder sets and natural codings for almost all translations on the two-dimensional torus. Due to the lack of a dominant eigenvector and the fact that we lose the self-similarity properties present for substitutive systems we cannot follow the known arguments from the substitutive case and have to develop new proofs to obtain our results in the $S$-adic setting.

Mots clés: S-adic shifts; substitutions; tilings; Pisot numbers; continued fractions; Brun algorithm; Arnoux-Rauzy algorithm; Lyapunov exponents

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

Valérie Berthé, Wolfgang Steiner, Jörg M. Thuswaldner. Geometry, dynamics, and arithmetic of $S$-adic shifts. 2016. ⟨hal-01070551v3⟩