Substitutions et pavages – SubTile

This project is aimed to the study of non-periodic tilings, particularly those tilings generated by local matching rules or substitutive rules. The discovery of the first example of a nonperiodic tiling of the plane goes back to the 60's. The knowledge on the domain hugely increased since. Let us mention the Penrose tiling, and the discovery of quasi-crystals. During the last 20 years, these objects have arisen a growing interest, with the interaction of several viewpoints: combinatorics, cohomology, ergodic theory, topology, algebra, noncommutative geometry, physics, computer science. The project gathers several recognized specialists of one or more of these fields, with notable contibutions in the domain of tilings. We have multiple, complementary and connected aims, which concern: the proof of new results, the creation of a software package to build and manipulate tilings, and compute invariants, and the diffusion of these aspects toward schools, universities, museums and websites. The software is an essential element for the diffusion of our results. This project includes several research directions. ' The Goodman-Strauss problem. In dimension 1, subshifts of finite type and substitutive subshifts form two disjoint families. In higher dimension, every substitutive tiling is a subshift of finite type; however, relations between the substitution and the local matching rules are at the moment not well understood, and not really explicit. The sofware package should be of a great help here. ' The discrete spectrum conjecture for Pisot substitutions : We will make a spectral study of tiling systems. There are a lot of work in dimension 1, but the discrete spectrum conjecture has only been solved for alphabet with 2 letters. Several equivalent forms of this conjecture are known (notably in terms of "coincidence"). Some of the work was generalized by Solomyak to substitutive tilings. In the Pisot multidimensional case, it would be interesting to find an other form for coincidences, and to give in that case a combinatorial version of the conjecture. In particular, one would like to have a mechanical way to decide whether a given tiling system has discrete spectrum. ' Existence and stability of quasi-crystal. Substitutive tilings have given for a longtime a model of the atomic structure of quasicrystals. From the combinatorial, ergodic or spectral properties of these tilings, one can deduce some physical properties of quasi-crystals. To give an example, the spectrum of the dynamical system associated with a tiling gives informations on the diffraction spectrum of the quasi-crystal. But we still do not understand really this correspondance : why does the fast cooling of some alloy produce a quasi-crystal' ' Explicit computation of cohomological invariants. Algorithms are known for the calculation of the cohomological invariants of substitution tilings of small dimension (up to 2) or for canoncial cut and project tiling in small codimension (up to codimension 3) but the tilings we are interested in here have fractal acceptance domain, and so new ideas are needed. ' Construction of spectral triples for tilings. Recently Bellissard and Pearson have proposed a spectral triple for Cantor sets equipped with an ultra metric. This applies to the transversal of a tiling and we aim to generalise it to the whole tiling algebra. The interest of a free tiling software package has been explained above. The production of images is essential in this domain to understand the tiling, and also for the diffusion of our results, to other mathematicians, or to students. The members of this project have strong capacities in terms of diffusion of knowledge. We aim to develop and structure these capacities, and share them with the general public. This can be done through conferences, workshops and visit, specially in the partner of this project.