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Flows near compact invariant sets. Part I

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Teixeira P.

Fundamenta Mathematicae

2013

tom 223

nr 3

225-272

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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1 Fundamenta Mathematicae

1 Teixeira P.

1 2013

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Flows near compact invariant sets. Part I