Geometric realization for substitution tilings

TL;DR

This paper constructs a geometric realization for substitution tilings with hyperbolic linear expansion, linking tiling dynamics to hyperbolic toral automorphisms and proposing a higher-dimensional Pisot Substitution Conjecture.

Contribution

It introduces a method to semi-conjugate substitution tiling dynamics to toral automorphisms using cohomology and formulates a higher-dimensional Pisot Substitution Conjecture.

Findings

Constructs a finite-to-one semi-conjugacy called geometric realization.

Shows conditions under which the geometric realization covers the entire torus.

Proposes a higher-dimensional generalization of the Pisot Substitution Conjecture.

Abstract

Given an n-dimensional substitution whose associated linear expansion is unimodular and hyperbolic, we use elements of the one-dimensional integer \v{C}ech cohomology of the associated tiling space to construct a finite-to-one semi-conjugacy, called geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If the linear expansion satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of the linear expansion, the image of geometric realization is the entire torus and coincides with the map onto the maximal equicontinuous factor of the translation action on the tiling space. We are led to formulate a higher-dimensional generalization of the Pisot Substitution Conjecture: If the linear expansion satisfies the Pisot family condition and the rank of the one-dimensional cohomology of the tiling space equals the generalized degree of the linear expansion, then the translation action on the tiling space has pure discrete spectrum.