If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space ${}_{\Phi }$ 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.

Suppose $A\in G{l}_d\left(\mathbb{Z}\right)$ has a 2-dimensional expanding subspace $E^u$, satisfies a regularity condition, called “good star”, and has $A^{*}\ge 0$, where $A^{*}$ is an oriented compound of $A$. A morphism $\theta$ of the free group on $\{1,2,\cdots ,d\}$ is called a non-abelianization of $A$ if it has structure matrix $A$. We show that there is a tiling substitution$\Theta$ whose “boundary substitution” $\theta =\partial \Theta$ is a non-abelianization of $A$. Such a tiling substitution $\Theta$ leads to a self-affine tiling of $E^u\sim {ℝ}^2$ with $A_u{:=A|}_{E_u}\in G{L}_2\left(ℝ\right)$ as its expansion. In the last section we find conditions on $A$ so...