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

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.

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)

References in corpus (5)

• Recognizability for sequences of morphisms
• Diffusion for chaotic plane sections of 3-periodic surfaces
• A combinatorial approach to products of Pisot substitutions
• The Pisot Conjecture for $\beta$-substitutions
• Some monoids of Pisot matrices

Cited by in corpus (15)

• Recognizability for sequences of morphisms
• Random Substitution Tilings and Deviation Phenomena
• Markov partitions for toral ${\mathbb{Z}}^2$-rotations featuring Jeandel-Rao Wang shift and model sets
• Rauzy induction of polygon partitions and toral ${\mathbb{Z}}^2$-rotations
• $\mathbf{S}$-adic sequences. A bridge between dynamics, arithmetic, and geometry
• Symbolic coding of linear complexity for generic translations of the torus, using continued fractions
• The Hoelder Property for the Spectrum of Translation Flows in Genus Two
• Arnoux-Rauzy interval exchange transformations
• On The Dimension Group of Unimodular S-Adic Subshifts
• Rauzy fractals of random substitutions
• Balancedness and coboundaries in symbolic systems
• On triangular paperfolding patterns
• The homology core and invariant measures
• The $S$-adic Pisot conjecture on two letters
• Absence of absolutely continuous diffraction spectrum for certain S-adic tilings