Propagation in media as a probe for topological properties

The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete classification of topological phases remains elusive. In this context, experimental studies are key, both for studying the interplay between topology and complex effects and for identifying new forms of topological order. It is therefore crucial to find complementary means to measure topological properties in order to reach a fundamental understanding of topological phases. In one dimensional chiral systems, we suggest a new way to construct and identify topologically protected bound states, which are the smoking gun of these materials. In two dimensional Hofstadter strips (i.e: systems which are very short along one dimension), we suggest a new way to measure the topological invariant directly from the atomic dynamics.