Mini-Workshop: The Pisot Conjecture - From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets

This mini-workshop brought together researchers with diverse backgrounds and a common interest in facets of the Pisot conjecture, which relates certain properties of a substitution to dynamical properties of the associated subshift. Mathematics Subject Classification (2000): 37B10, 28A80, 37B50, 52C23. Introduction by the Organisers A substitution is a non-erasing morphism of the free monoid. Subshifts generated by fixed points of substitutions are natural symbolic models for deterministic self-similar dynamical systems. The Pisot conjecture relates number theoretic properties of the substitution matrix to dynamical properties of the generated subshift. Explicitly, it states that the symbolic dynamical system of a unimodular Pisot substitution has pure point spectrum. This conjecture has attracted a fair amount of attention. In fact, Pisot substitutions systems and the Pisot conjecture have numerous applications, for example to Diophantine approximation, equidistribution properties of toral translations and low discrepancy sequences, beta-shifts, multidimensional continued fraction expansions, generation or recognition of arithmetic discrete planes, or else effective construction of Markov partitions for toral automorphisms, the main eigenvalue of which is a Pisot number. Furthermore, the conjecture is supported by numerical evidence since it can be reformulated in effective terms. Still, so far it has only been proved in the case of two symbols. 726 Oberwolfach Report 13 Primitive substitutions can not only be studied in the framework of symbolic dynamics but also in a higher dimensional geometric setting. There, one is interested in substitution-generated tilings and Delone sets. In this situation, there is an analogous version of the Pisot conjecture. There exist several necessary and/or sufficient conditions for pure point spectrum for substitution dynamical systems. In fact, three related approaches to pure point spectrum have been developed in the last twenty years: One approach is based on the notion of coincidence, introduced by Dekking, then by Host, in an unpublished paper, and lastly in greater generality by Arnoux and Ito and Hollander and Solomyak. This correspondence is especially apparent in the recent work of Barge and Kwapisz who showed that pure point spectrum is equivalent to what they call the geometric coincidence condition. This condition is algorithmically decidable. A different approach relies on the geometric representation of substitution dynamical systems with pure point spectrum as translations on compact metric groups such as shown by the pioneering work of Rauzy in the 80’s on the socalled Rauzy fractal. The Pisot conjecture can then be translated in tiling terms. The geometric coincidence can also be stated in this framework. Finally, there is an approach connecting pure point spectrum with cut and project schemes and so-called Meyer sets. In a very recent work dealing with the higher dimensional case, Lee has shown that a primitive substitution Delone set has pure point spectrum if and only if it comes from a cut and project scheme. Thus, Lee’s characterization links pure point spectrum and cut and project schemes within the framework of primitive substitutions. A crucial ingredient in her proof is a new coincidence condition for Delone sets generalizing all earlier conditions of this kind in the geometric setting. Another important ingredient is her recent work with Solomyak showing that pure point diffraction implies the Meyer property for primitive substitution systems, thereby answering a question of Lagarias. This is then combined with a new understanding of cut and project schemes in terms of topologies brought forward in recent work of Baake and Moody. Mini-Workshop: The Pisot Conjecture 727 Mini-Workshop: The Pisot Conjecture From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets