The branch locus for one-dimensional Pisot tiling spaces

Volume 204 / 2009

Marcy Barge, Beverly Diamond, Richard Swanson Fundamenta Mathematicae 204 (2009), 215-240 MSC: Primary 37B05; Secondary 37A30, 37B50, 54H20. DOI: 10.4064/fm204-3-2

Abstract

If $\varphi$ is a Pisot substitution of degree $d$, then the inflation and substitution homeomorphism $\mit\Phi$ on the tiling space ${\cal T}_{\mit\Phi}$ factors via geometric realization onto a $d$-dimensional solenoid. Under this realization, the collection of $\mit\Phi$-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.

Authors

  • Marcy BargeDepartment of Mathematics
    Montana State University
    Bozeman, MT 59717, U.S.A.
    e-mail
  • Beverly DiamondDepartment of Mathematics
    College of Charleston
    Charleston, SC 29424, U.S.A.
    e-mail
  • Richard SwansonDepartment of Mathematics
    Montana State University
    Bozeman, MT 59717, U.S.A.
    e-mail

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