Flows near Compact Invariant Sets - Part I

Abstract

In this paper it is proved that near a compact, invariant, proper subset of a continuous flow on a compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. This result shows that assuming the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour, in the vicinity of compact invariant sets, than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings to light, in a systematic way, the possibility of occurrence of orbits of infinite height arbitrarily near the compact invariant in question, and this under relatively simple conditions. Singularities of smooth vector fields displaying this strange phenomenon occur in every dimension greater than 2 (in this paper, a smooth flow on the 3-dimensional sphere exhibiting such an equilibrium is constructed). Near periodic orbits, the same phenomenon is observable already in dimension 4 (and on every manifold of dimension greater than 4). As a corollary to the main result, an elegant characterization of the topological Hausdorff structure of the set of all compact minimal sets of the flow is obtained (Theorem 2). Keywords: topological behaviour of C0 flows, compact invariant sets, compact minimal sets, topological Hausdorff structure, non-hyperbolic singularities and periodic orbits, orbits of infinite height.