Geometry, dynamics, and arithmetic of $S$-adic shifts
Abstract
This paper studies geometric and spectral properties of -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 -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 -adic framework. They are applied to families of -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 -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 -adic setting.
Cite
@article{arxiv.1410.0331,
title = {Geometry, dynamics, and arithmetic of $S$-adic shifts},
author = {Valérie Berthé and Wolfgang Steiner and Jörg Thuswaldner},
journal= {arXiv preprint arXiv:1410.0331},
year = {2020}
}
Comments
After this paper was published in Ann. Inst. Fourier (Grenoble) (2019), we observed that Theorem 3.3 holds in a more general setting. The present manuscript contains this more general version of Theorem 3.3. To prove it we needed to change the last part of its proof (see p.32)