Geometric recognisability for FLC patterns

Abstract: Informally, recognisability of a substitution (or ‘inflate, subdivide’) rule regards its invertibility. Beyond the classical result of Mossé, highly general recognisability results have recently been established in the symbolic setting, even going beyond substitutions to S-adic sequences. In this talk, I will introduce a notion of a geometric tiling, point set or generalised ‘pattern’ of Euclidean space being substitutional with respect to an inflation map, in terms of two basic relations from Aperiodic Order: local derivability and local indistinguishability. These are analogues, from Symbolic Dynamics, of being related by sliding block codes and being in the same orbit closure, respectively. In the translational finite local complexity (FLC) case (e.g., tilings with finitely many tile types, meeting in finitely many ways up to translation), we give a formula for the number of pre-images of a pattern under substitution in terms of its group of translational periods. In particular, for a suitable power of substitution, the non-periodic tilings are precisely those with unique pre-images under substitution. Many techniques used are similar to those in Solomyak’s unique composition result, although we require no minimality (primitivity) assumption. If time permits, I will explain the motivation for this work, on the question of when a cut and project scheme produces substitutional patterns.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

---

One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

---