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Let $\varphi$ be a substitution of Pisot type on the alphabet $𝒜=\{1,2,...,d\}$; $\varphi$ satisfies theif for every $i,j\in 𝒜$, there are integers $k,n$ such that ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same $k$-th letter, and the prefixes of length $k-1$ of ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d=2$ and provide a partial result for $d\ge 2$.

We present a new technique for showing that inverse limit spaces of certain one-dimensional Markov maps are not homeomorphic. In particular, the inverse limit spaces for the three maps from the tent family having periodic kneading sequence of length five are not homeomorphic.

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space ${}_{\phi }$ has pure discrete spectrum, we describe the collection ${}_{\phi }^P$ of pairs of proximal tilings in ${}_{\phi }$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then ${}_{\phi }$ and ${}_{\psi }$ are homeomorphic if and...

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.