Asymptotic structure in substitution tiling spaces

TL;DR

This paper investigates the asymptotic structure of substitution tiling spaces, introducing the concept of the branch locus to understand asymptotic behavior and proving its invariance under certain conditions, with specific results for 2D Pisot tilings.

Contribution

It defines the branch locus as a topological invariant for substitution tiling spaces and characterizes it for 2D Pisot cases, advancing understanding of asymptotic tiling structures.

Findings

Branch locus is a topological invariant under certain asymptotic conditions.

In 2D Pisot substitution tilings, the branch locus is an inverse limit of an expanding Markov map.

Every regular tiling space has asymptotic tilings in some directions.

Abstract

Every sufficiently regular space of tilings of $\R^d$ has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open $(d-1)$-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity `starts'. This leads to the definition of the {\em branch locus} of the tiling space: this is a subspace of the tiling space, of dimension at most $d-1$, that summarizes the `asymptotic in at least a half-space' behavior in the tiling space. We prove that if a $d$-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed $(d-1)$-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a 2-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a 1-dimensional simplicial complex.