Detecting invariant manifolds of dynamical systems using persistent homology

We use methods of Persistent Homology Theory to study invariant manifolds of dynamical systems. We first establish connections between the persistence diagrams of two sets which are close to each other, with respect to the Hausdorff distance. We then apply these results to study properties of limit sets of specific dynamical systems, by using the persistence diagram of a numerically obtained sample set. Under mild assumptions, we show how to use numerical data to state analytical results concerning the geometry of the limit sets.