Geometry, dynamics, and arithmetic of S-adic shifts
Annales de l'Institut Fourier, to appear, 63 p.

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. 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. It is shown that almost all of these shifts have 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 self-similarity properties present for substitutive systems we have to develop new proofs to obtain our results in the S-adic setting.

Cet article étudie les propriétés géométriques et spectrales de décalages S-adiques engendrés par fractions continues. Ces systèmes dynamiques symboliques sont obtenus par itération adique d’une suite de substitutions. Nous montrons que ces décalages sont à spectre purement discret et étudions les propriétés des ensembles fractals de Rauzy associés sous une hypothèse de type Pisot généralisée ainsi qu’une condition géométrique de coïncidence. Ces résultats étendent la portée de la conjecture Pisot substitutive au cadre S-adique. Nous montrons que presque tous les décalages d’Arnoux–Rauzy ont un spectre purement discret. En utilisant des mots S-adiques liés à l’algorithme de fraction continue de Brun, nous exhibons des ensembles à restes bornés et des codages symboliques pour presque toutes les translations du tore bidimensionnel. En raison de l’absence des propriétés d’autosimilarité des systèmes substitutifs, nous devons développer de nouvelles preuves dans le cadre S-adique.

Received : 2016-08-29
Revised : 2018-05-30
Accepted : 2018-06-12
Classification:  37B10,  37A30,  11K50,  28A80
Keywords: Symbolic dynamics, non-stationary dynamics, S-adic shifts, substitutions, tilings, Pisot numbers, continued fractions, Brun algorithm, Arnoux–Rauzy algorithm, Lyapunov exponents
     author = {Berth\'e, Val\'erie and Steiner, Wolfgang and Thuswaldner, J\"org M.},
     title = {Geometry, dynamics, and arithmetic of $S$-adic shifts},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Berthé, Valérie; Steiner, Wolfgang; Thuswaldner, Jörg M. Geometry, dynamics, and arithmetic of $S$-adic shifts. Annales de l'Institut Fourier, to appear, 63 p.

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