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A complete invariant for the topology of one-dimensional substitution tiling spaces

Published online by Cambridge University Press:  02 October 2001

MARCY BARGE  and
BEVERLY DIAMOND
MARCY BARGE
Affiliation:
Department of Mathematics, Montana State University, Bozeman, MT 59717, USA (e-mail: [email protected])
BEVERLY DIAMOND
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA (e-mail: [email protected])
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Abstract

Let \varphi be a primitive, non-periodic substitution. The tiling space \mathcal{T}_\varphi has a finite (non-zero) number of asymptotic composants. We describe the form and make use of these asymptotic composants to define a closely related substitution \varphi^* and prove that for primitive, non-periodic substitutions \varphi and \chi, \mathcal{T}_\varphi and \mathcal{T}_\chi are homeomorphic if and only if {\varphi^*} (or its reverse) and \chi^* are weakly equivalent. We also provide examples indicating that for substitution minimal systems, flow equivalence and orbit equivalence are independent.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 21 , Issue 5 , October 2001 , pp. 1333 - 1358
Copyright
2001 Cambridge University Press

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