Embedding tiling spaces in surfaces

Volume 201 / 2008

Charles Holton, Brian F. Martensen Fundamenta Mathematicae 201 (2008), 99-113 MSC: Primary 37B50; Secondary 37C70, 37E35, 37B10, 37E05. DOI: 10.4064/fm201-2-1

Abstract

We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classification is nearly complete. The results characterize the codimension one hyperbolic attractors of surface diffeomorphisms in terms of asymptotic composants of substitutions.

Authors

  • Charles HoltonDepartment of Mathematics
    The University of Texas at Austin
    1 University Station//C1200
    Austin, TX 78712, U.S.A.
    e-mail
  • Brian F. MartensenMinnesota State University
    Wissink 273
    Mankato, MN 56001, U.S.A.
    e-mail

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