The branch locus for one-dimensional Pisot tiling spaces
Volume 204 / 2009
Fundamenta Mathematicae 204 (2009), 215-240
MSC: Primary 37B05; Secondary 37A30, 37B50, 54H20.
DOI: 10.4064/fm204-3-2
Abstract
If $\varphi$ is a Pisot substitution of degree $d$, then the inflation and substitution homeomorphism $\mit\Phi$ on the tiling space ${\cal T}_{\mit\Phi}$ factors via geometric realization onto a $d$-dimensional solenoid. Under this realization, the collection of $\mit\Phi$-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.
