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

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.

Augmenting dimension group invariants for substitution dynamics

“…This element is an invariant of isomorphism of such algebras, so according to Theorem 4.3 we have that (K 0 (X τ ), [1]) is an invariant of one-sided conjugacy of π + (X τ ).…”

“…This pair of examples is also resistant to the method of comparing the configuration of the special elements or asymptotic orbits (cf. [1]), suggested to us by Charles Holton. Indeed, the "configuration data" of all right or left tail equivalence classes of special elements (see [10]) are identical.…”

“…The K 0 -group associated to a unital C * -algebra posseses a distinguished element [1] corresponding to the unit of the C * -algebra. This element is an invariant of isomorphism of such algebras, so according to Theorem 4.3 we have that (K 0 (X τ ), [1]) is an invariant of one-sided conjugacy of π + (X τ ).…”

“…From [1, 3.10] τ and υ are flow equivalent precisely when the derived substitutions τ * and υ * defined there are weakly equivalent. However, since computer experiments indicate that both τ * and υ * allow squares ww but no triples www in their respective languages, the method given in [1] for establishing flow inequivalence does not seem to work here. Nevertheless, we can use our invariant to prove that the shift spaces associated to these two substitutions are not flow equivalent.…”

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \beta -shifts

“…(4) As a consequence, if φ is Pisot then it is automatically translationally aperiodic if dim(V ) 2.…”