Self-similarity and limit spaces of substitution tiling semigroups

2021/12/14 by James Walton, Walton, James J., Michael F. Whittaker +1
Materials Science · Mathematics · #52C22 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary: 37B52 #Quasicrystal Structures and Properties #Secondary: 20M18

paper · pdf · doi:10.48550/arxiv.2112.07652

openalex publication_date 2021/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that Kellendonk's tiling semigroup of an FLC substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson--Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.

Related

• From Self-Similar Structures to Self-Similar Groups
• Forward limit sets of semigroups of substitutions
• Homeomorphisms of One-dimensional Inverse Limits with Applications to Substitution Tilings, Unstable Manifolds, and Tent Maps
• Self-similar groups and topological dimension
• Second Order Ergodic Theorem for Self-Similar Tiling Systems