Flows near compact invariant sets. Part I

Pedro Teixeira. "Flows near compact invariant sets. Part I." Fundamenta Mathematicae 223.3 (2013): 225-272. <http://eudml.org/doc/282689>.

@article{PedroTeixeira2013,
abstract = {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 ³ 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).},
author = {Pedro Teixeira},
journal = {Fundamenta Mathematicae},
keywords = {topological behaviour of flows; compact invariant sets; compact minimal sets; topological Hausdorff structure; non-hyperbolic singularities and periodic orbits; orbits of infinite height},
language = {eng},
number = {3},
pages = {225-272},
title = {Flows near compact invariant sets. Part I},
url = {http://eudml.org/doc/282689},
volume = {223},
year = {2013},
}

TY - JOUR
AU - Pedro Teixeira
TI - Flows near compact invariant sets. Part I
JO - Fundamenta Mathematicae
PY - 2013
VL - 223
IS - 3
SP - 225
EP - 272
AB - 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 ³ 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).
LA - eng
KW - topological behaviour of flows; compact invariant sets; compact minimal sets; topological Hausdorff structure; non-hyperbolic singularities and periodic orbits; orbits of infinite height
UR - http://eudml.org/doc/282689
ER -