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Abstract

It is proved 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. Theorem 1 shows that 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 set in question, and this under relatively simple conditions. Singularities of $C^{\infty }$ vector fields displaying this strange phenomenon occur in every dimension $n\geq 3$ (in this paper, a $C^{\infty }$ flow on $\mathbb {S}^{3}$ exhibiting such an equilibrium is constructed). Near periodic orbits, the same phenomenon is observable in every dimension $n\geq 4$. As a corollary to the main result, an elegant characterization of the topological-dynamical Hausdorff structure of the set of all compact minimal sets of the flow is obtained (Theorem 2).