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Proximality in Pisot tiling spaces

100%
Barge M., Diamond B.
Fundamenta Mathematicae
|
2007
|
tom 194
|
nr 3
191-238
EN
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 $𝓣_{φ}$ has pure discrete spectrum, we describe the collection $𝓣^{P}_{φ}$ of pairs of proximal tilings in $𝓣_{φ}$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then $𝓣_{φ} $ and $𝓣_{ψ}$ are homeomorphic if and only if $𝓣^{P}_{φ}$ and $𝓣^{P}_{ψ}$ are homeomorphic. We make use of this invariant to distinguish tiling spaces for which other known invariants are ineffective. In addition, we show that for strong Pisot substitutions, pure discrete spectrum of the flow on the associated tiling space is equivalent to proximality being a closed relation on the tiling space.
2
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Homeomorphisms of inverse limit spaces of one-dimensional maps

100%
Barge M., Diamond B.
Fundamenta Mathematicae
|
1994-1995
|
tom 146
|
nr 2
171-187
EN
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.
3
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The branch locus for one-dimensional Pisot tiling spaces

81%
Barge M., Diamond B., Swanson R.
Fundamenta Mathematicae
|
2009
|
tom 204
|
nr 3
215-240
EN
If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space $𝓣_{Φ}$ 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.
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