Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of \nats possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech compactification \beta \nats . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225, arXiv:1301.511

Maximal equicontinuous factors and cohomology for tiling spaces

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that this map is injective in degree one and has torsion free cokernel. We show by example, however, that the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology

Homological Pisot Substitutions and Exact Regularity

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.Comment: 16 pages, LaTeX, no figure