The Pisot Conjecture

1/9/14 - 5/9/14 Lorentz Centre, International Centre for workshops in the sciences, Leiden, The Netherlands

Scientific Report on “The Pisot Conjecture”

Henk Bruin, Alex Clark, Robbert Fokkink

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 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 they are the same if the inflation factor is a 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 was centred around four main speakers, all of whom gave 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.

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.

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