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

TL;DR

This paper explores the geometric and spectral properties of $S$-adic shifts, extending the Pisot substitution conjecture, and demonstrates that most such systems have pure discrete spectrum with applications to toral translations.

Contribution

It generalizes the Pisot substitution conjecture to the $S$-adic framework and develops new methods to analyze their spectral and geometric properties.

Findings

Most $S$-adic shifts generated by Arnoux-Rauzy and Brun substitutions have pure discrete spectrum.

Established tiling properties of Rauzy fractals under generalized Pisot assumptions.

Constructed bounded remainder sets and codings for translations on the 2-torus.

Abstract

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.