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


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.


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

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