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

Let phi be a primitive, non-periodic substitution. The tiling space T-phi 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 phi* and prove that for primitive, non-periodic substitutions phi and chi, T-phi and T-chi are homeomorphic if and only if phi* (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.