If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-3-2,
author = {Marcy Barge and Beverly Diamond and Richard Swanson},
title = {The branch locus for one-dimensional Pisot tiling spaces},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {215-240},
zbl = {1185.37013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-3-2}
}
Marcy Barge; Beverly Diamond; Richard Swanson. The branch locus for one-dimensional Pisot tiling spaces. Fundamenta Mathematicae, Tome 205 (2009) pp. 215-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-3-2/