Central sets and substitutive dynamical systems

TL;DR

This paper links central sets and the strong coincidence conjecture for Pisot substitutions, offering a new arithmetical perspective on the Pisot substitution conjecture through ultrafilters and combinatorics.

Contribution

It establishes a novel connection between central sets and the strong coincidence conjecture, reformulating the conjecture in terms of ultrafilters and central sets.

Findings

Reformulation of the strong coincidence condition using central sets.

New approach to the Pisot substitution conjecture via ultrafilters.

Integration of combinatorics on words, tilings, and topological dynamics.

Abstract

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 $\nats.$