A complete invariant for the topology of one-dimensional substitution tiling spaces (Q2757007)

Any primitive, non-periodic substitution \(\phi\) has an associated tiling space \(T_{\phi}\): this is the topological space of all possible tilings using distinct intervals for each symbol in \(\phi\) that mimic the pattern of symbols in an infinite word invariant under \(\phi\), and with two tilings close if they agree on a large region after a small translation. Here the asymptotic composants of \(T_{\phi}\) are described and used to give a new invariant of the tiling spaces: \(T_{\phi}\) and \(T_{\chi}\) are homeomorphic if and only if certain associated substitutions \(\phi^*\) and \(\chi^*\) are (possibly up to reversal) weakly equivalent.