Geometric theory of unimodular Pisot substitutions

Abstract

We are concerned with the tiling flow T associated to a substitution φ over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue λ > 1. The motivation is provided by the (still open) conjecture asserting that T has pure discrete spectrum for any such φ. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual R d -1 -action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as sufficient conditions only-but they all emerge from a unified approach based on the new device: the strand space F φ of φ. The proof of the necessity hinges on determination of the discrete spectrum of T as that of the associated Kronecker toral flow.

Non-Unimodularity

“…One is often (or usually) interested whether the sets Ω j are the prototiles of a specific aperiodic tiling since this implies that the underlying dynamical system is pure point, see the remarks in the talk by Akiyama [1] and compare [5,8,13]. This condition can be checked algorithmically by making use of the above mentioned measure-disjointness on the right-hand side of the associated IFS, compare [12] (also see [13,Sections 6.9 & 6.10]).…”

“…This condition can be checked algorithmically by making use of the above mentioned measure-disjointness on the right-hand side of the associated IFS, compare [12] (also see [13,Sections 6.9 & 6.10]). Another way to phrase this is by so-called coincidence conditions and/or (weak) finiteness conditions, see Akiyama's talk [1] and [5,8,13] and references therein.…”

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \beta -shifts

“…In case the abelianization of φ is unimodular, has irreducible characteristic polynomial, and has dominant eigenvalue a Pisot-Vijayaraghavan number (the "irreducible, unimodular Pisot" case), the geometric theory of the tiling flow alluded to above is developed in [5]. Here we extend the theory to cover all primitive substitutions of Pisot type.…”

“…The substitution based geometric approach initiated by Rauzy has been developed by Arnoux and Ito, and Cantorini and Siegel ( [3,9]), and recast from the Iterated Function Systems point of view by Sirvent and Wang ([26]). Further advances were made independently in [16] and [5] where an optimal coincidence condition in the irreducible unimodular case was introduced. The optimality alludes here to equivalence with various good properties ranging from some very specific tiling and metric properties of the (generalized) Rauzy fractals to the general measure theoretical property of pure discrete spectrum (cf.…”

“…The optimality alludes here to equivalence with various good properties ranging from some very specific tiling and metric properties of the (generalized) Rauzy fractals to the general measure theoretical property of pure discrete spectrum (cf. [27]) of the tiling flow; see [5] for a comprehensive discussion. The related number-theoretic investigations have been undertaken in a number of works by Akiyama, Frougny, Ito, Rao, Solomyak, Steiner, and Thuswaldner( [1,2,13,16,29]).…”