Topological Measurements of Invariant Sets in Discrete Dynamical Systems

Coarse, topological measurements of dynamical systems defined by maps can be used to uncover information about invariant sets for the system.  These techniques are based on the Conley index and have been used to detect invariant structures from fixed points and periodic orbits, to connecting orbits and sets which exhibit chaotic symbolic dynamics.  More notably, these techniques have been used to study both finite-dimensional and infinite-dimensional systems and to prove the existence of unstable invariant sets. The method relies on first building a pair of compact sets called an index pair and then computing the relative homology of the pair.  This information is then used to make conclusions about the associated invariant structures.