The Branch Locus for One-Dimensional Pisot Tiling Spaces

TL;DR

This paper investigates the structure of tiling spaces generated by Pisot substitutions, revealing how their branch loci in the torus are invariant under homeomorphisms, thus linking topological properties to algebraic and geometric features.

Contribution

It establishes a relationship between homeomorphic Pisot tiling spaces and their branch loci, showing invariance up to affine transformations in the torus.

Findings

Branch loci are finite sets projecting onto the d-torus.

Homeomorphic tiling spaces have equivalent branch loci up to affine maps.

The geometric realization connects tiling dynamics with solenoid and torus structures.

Abstract

If phi is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Phi on the tiling space T_Phi factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Phi-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.