Geometry, dynamics, and arithmetic of S-adic shifts

Jörg Thuswaldner
Montanuniversität Leoben, Austria

Date(s) : 06/10/2017   iCal
10 h 00 min - 16 h 00 min

This talk 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.



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