Flows near compact invariant sets. Part I

Pedro Teixeira¹

¹ Centro de Matemática da Universidade do Porto Rua do Campo Alegre, 687 4169-007 Porto, Portugal

Fundamenta Mathematicae, Tome 223 (2013) no. 3, pp. 225-272.

Résumé

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).

Détail

DOI : 10.4064/fm223-3-3

Keywords: proved near compact invariant proper subset flow locally compact connected metric space least out twenty eight relevant dynamical phenomena necessarily occur theorem shows connectedness phase space implies existence considerably deeper classification topological flow behaviour vicinity compact invariant sets described classical theorems ura kimura bhatia proposed classification brings light systematic possibility occurrence orbits infinite height arbitrarily near compact invariant set question under relatively simple conditions singularities infty vector fields displaying strange phenomenon occur every dimension geq paper infty flow mathbb exhibiting equilibrium constructed near periodic orbits phenomenon observable every dimension geq corollary main result elegant characterization topological dynamical hausdorff structure set compact minimal sets flow obtained theorem