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2013 | 223 | 3 | 225-272
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Flows near compact invariant sets. Part I

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It is proved that near a compact, invariant, proper subset of a C⁰ 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 $C^{∞}$ vector fields displaying this strange phenomenon occur in every dimension n ≥ 3 (in this paper, a $C^{∞}$ flow on 𝕊³ exhibiting such an equilibrium is constructed). Near periodic orbits, the same phenomenon is observable in every dimension n ≥ 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).
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  • Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
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