The Branch Locus for One-dimensional Pisot Tiling Spaces
نویسنده
چکیده
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 the resulting branch loci are the same up to the action of certain affine maps on the torus. 2000 Mathematics Subject Classification: Primary: 37B05; Secondary: 37A30, 37B50, 54H20
منابع مشابه
Homological Pisot Substitutions and Exact Regularity
We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and the first rational Čech cohomology is d-dimensional. We construct examples of such “homological Pisot” substitutions whose tiling flows do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide ...
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For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion β is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion β, such that no tiling admitted by the substitution is locally finite. The proofs...
متن کاملar X iv : m at h / 05 06 09 8 v 1 [ m at h . D S ] 6 J un 2 00 5 GENERALIZED β - EXPANSIONS , SUBSTITUTION TILINGS , AND LOCAL FINITENESS
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion β is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion β, such that no tiling admitted by the substitution is locally finite. The proofs...
متن کاملar X iv : m at h / 05 06 09 8 v 2 [ m at h . D S ] 8 N ov 2 00 5 GENERALIZED β - EXPANSIONS , SUBSTITUTION TILINGS , AND LOCAL FINITENESS
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion β is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion β, such that no tiling admitted by the substitution is locally finite. The proofs...
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