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 (any dimension) 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 relation by sliding block code 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, non-periodic patterns are precisely those with unique pre-images under substitution. Several techniques used are similar to those in Solomyak’s unique composition result, although all minimality (primitivity) requirements are dropped. If time permits, I will explain the motivation for this work, on the question of when a cut and project scheme produces substitutional patterns.

Date
Location
IRIF, Paris