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The Pisot Conjecture
The Pisot (substitution) conjecture has been lying around for several decades now. The true origin of the conjecture remains unknown, and perhaps there does not exist one true origin, as often happens with good conjectures. It emerged from studies in computer science, math, and physics. By coincidence, almost at the same time of this workshop, a paper by Pierre Arnoux and Edmund Harris appeared in the Notices of the AMS, presenting the main unsolved problem in the study of aperiodic order as
The Pisot conjecture: The one-dimensional tiling generated by a Pisot substitution rule on d letters with determinant ±1 is a cut-and-project quasicrystal.
This is not a conjecture that is easily stated in the standard terminology of a working mathematician, but the combination of the words “one-dimensional”, “letters”, “quasicrystal”, indicate that something multidisciplinary is in the offing. There are other ways to state the conjecture. In a recent text on this topic - Mathematics of Aperiodic Order (2015, Birkhäuser) – the quasicrystal is replaced by a system of pure discrete spectrum, i.e., a translation on a compact abelian group.
The Pisot conjecture predicts that certain tilings and certain symbolic sequences are equivalent, if put in the context of dynamical systems. Tilings and infinite words are self-similar structures, which are generated by an inflation rule. The archetypes are the Penrose tiling and the Thue-Morse word. The conjecture predicts that a tiling and a word are equivalent if their inflation factor is the same Pisot number.
The conjecture remains unsolved, but the gap between theoretical and computational studies is closing. On the theoretical side, Marcy Barge has recently proved that the conjecture is true for Parry numbers, which form a significant subset of the Pisot numbers. On the computational side, Franz Gähler has checked the conjecture for hundreds of thousands Pisot number of small degree. Both of them reported on their recent progress on the conjecture.
The workshop centred around four main speakers, who delivered a series of lectures. Michael Baake talked about quasicrystals and harmonic analysis. Marcy Barge talked about his recent progress on the conjecture. Fabien Durand talked about the spectrum of minimal Cantor sets. Klaus Schmidt talked about algebraic dynamics on compact abelian groups. Several other participants, all specialists on aperiodic order, reported their latest results in this area: Shigeki Akiyama, Jarek Kwapisz, Lorenzo Sadun, Wolfgang Steiner, Luca Zamboni. The contents of many of these talks, and other contributions by participants, appear in written form in a special issue of Topology and its Applications:
· Michael Baake, Franz Gaehler - Pair correlations of aperiodic inflation rules via renormalisation: some interesting examples
· Marcy Barge - Geometric and spectral properties of Pisot substitutions
· Dan Rust - An uncountable set of tiling spaces with distinct cohomology
· Lorenzo Sadun - Finitely balanced sequences and platicity of 1-dimensional tilings
· Klaus Schmidt - Representations of toral automorphisms
· Valerie Berthé, Milton Minervino, Wolfgang Steiner, Joerg Thuswaldner - The S-adic Pisot conjecture on two letters
· Martijn de Vries, Vilmos Komornik, Paola Loreti -Topology of Univoque bases
The Pisot conjecture has not been solved yet, and follow up workshops of this Lorentz Center meeting have already taken place or are being scheduled in Leicester, Galway, Lyon and Delft. We thank the Lorentz Center for its hospitality, and the Leverhulme Foundation for additional funding. We hope to return soon, once the conjecture is fully solved.
Henk Bruin (Vienna, Austria)
Alex Clark (Leicester, United Kingdom)
Robbert Fokkink (Delft, Netherlands)