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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
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 - 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
- ·
Marcy Barge - ·
Dan Rust - ·
Lorenzo Sadun
- ·
Klaus Schmidt - ·
Valerie Berthé,
Milton Minervino, Wolfgang Steiner, Joerg Thuswaldner - ·
Martijn de Vries, Vilmos Komornik, Paola Loreti - 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) [Back] |