Homological Pisot Substitutions and Exact Regularity

TL;DR

This paper studies a special class of one-dimensional substitution tiling spaces with Pisot dilatations, constructing examples that lack pure discrete spectra and revealing a new Exact Regularity Property that constrains their measure-theoretic structure.

Contribution

The authors introduce and analyze homological Pisot substitutions, providing the first examples without pure discrete spectra and establishing the ERP as a key property.

Findings

Constructed non-unimodular homological Pisot substitutions without pure discrete spectra.

Established the Exact Regularity Property (ERP) for these substitutions.

Conjectured that the coincidence rank divides a power of the dilatation's norm.

Abstract

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.