The branch locus for one-dimensional Pisot tiling spaces

The Branch Locus for One-dimensional Pisot Tiling Spaces

If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space TΦ 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 th...

Proximality in Pisot Tiling Spaces

A substitution φ is strong Pisot if its abelianization matrix is non-singular 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 Tφ has pure discrete spectrum, we describe the collection T P φ of pairs of proximal tilings in Tφ in a natural way as a s...

Cohomology in One-dimensional Substitution Tiling Spaces

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which “forces its border.” One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomor...

Tiling spaces, codimension one attractors and shape

We establish a close relationship between, on the one hand, expanding, codimension one attractors of diffeomorphisms on closed manifolds (examples of so-called strange attractors), and, on the other, spaces which arise in the study of aperiodic tilings. We show that every such orientable attractor is homeomorphic to a tiling space of either a substitution or a projection tiling, depending on it...

Self aÆne tiling and Pisot numeration system