Abstract
We show that an aperiodic minimal tiling space with only nitely 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 condi- tions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classication is nearly complete. The results charac- terize the codimension one hyperbolic attractors of surface dieomorphisms in terms of asymptotic composants of substitutions.
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