Overlapping substitutions and tilings

TL;DR

This paper extends the concept of substitution rules to include overlapping substitutions, analyzing their properties, ergodicity, and conditions for consistency, with applications to tilings and Delone sets.

Contribution

It introduces a framework for overlapping substitutions, linking substitution matrices to patch frequencies and ergodic properties, and provides conditions for their consistency.

Findings

Patch frequencies are uniformly convergent.

The associated expansion constant is an algebraic integer.

Conditions for avoiding illegal overlaps are established.

Abstract

We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector encodes the patch frequency of the resulting tiling. The patch frequencies are shown to be uniformly convergent, implying that the corresponding dynamical system is uniquely ergodic. Under mild assumptions, we further prove that the associated expansion constant is always an algebraic integer. In general, overlapping substitutions may yield a patch with illegal (partial) overlaps of tiles, even if it is locally consistent. We provide a sufficient condition for an overlapping substitution to be consistent, ensuring that no such illegal tiles emerge. Finally, we construct many intriguing one-dimensional overlapping substitutions and present higher dimensional examples from Delone multi-sets with inflation symmetry.