Embedding tiling spaces in surfaces
Volume 201 / 2008
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.
