Flows near compact invariant sets. I (Q2866777)

In this work, the author proves that near a compact, invariant, proper subset of a \(C^0\) flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. A first result proves that the connectedness of the phase space implies the existence of a considerably deeper classification of their topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. Near periodic orbits, the same phenomenon is observable in every dimension greater than three. As a corollary to the main result, the author obtains a characterization of the topological-dynamical Hausdorff structure of the set of all compact minimal sets of the flow.