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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm201-2-1,
author = {Charles Holton and Brian F. Martensen},
title = {Embedding tiling spaces in surfaces},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {99-113},
zbl = {1155.37016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm201-2-1}
}
Charles Holton; Brian F. Martensen. Embedding tiling spaces in surfaces. Fundamenta Mathematicae, Tome 201 (2008) pp. 99-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm201-2-1/