An attraction result and an index theorem for continuous flows on ℝ n × [ 0 , ∞ )

Klaudiusz Wójcik

Annales Polonici Mathematici (1997)

  • Volume: 65, Issue: 3, page 203-211
  • ISSN: 0066-2216

Abstract

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We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on E = ℝ n + 1 for which ∂ E = ℝ n × 0 is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ℝ n × [ 0 , ∞ ) .

How to cite

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Klaudiusz Wójcik. "An attraction result and an index theorem for continuous flows on $ℝ^n × [0,∞)$." Annales Polonici Mathematici 65.3 (1997): 203-211. <http://eudml.org/doc/269989>.

@article{KlaudiuszWójcik1997,
abstract = {We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^\{n+1\}$ for which $∂E = ℝ^n × \{0\}$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^n × [0,∞)$.},
author = {Klaudiusz Wójcik},
journal = {Annales Polonici Mathematici},
keywords = {Conley index; fixed point index; permanence},
language = {eng},
number = {3},
pages = {203-211},
title = {An attraction result and an index theorem for continuous flows on $ℝ^n × [0,∞)$},
url = {http://eudml.org/doc/269989},
volume = {65},
year = {1997},
}

TY - JOUR
AU - Klaudiusz Wójcik
TI - An attraction result and an index theorem for continuous flows on $ℝ^n × [0,∞)$
JO - Annales Polonici Mathematici
PY - 1997
VL - 65
IS - 3
SP - 203
EP - 211
AB - We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^{n+1}$ for which $∂E = ℝ^n × {0}$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^n × [0,∞)$.
LA - eng
KW - Conley index; fixed point index; permanence
UR - http://eudml.org/doc/269989
ER -

References

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  10. [Mr3] M. Mrozek, Open index pairs, the fixed point index and rationality of zeta function, Ergodic Theory Dynam. Systems 10 (1990), 555-564. 
  11. [Mr4] M. Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (1989), 177-194. Zbl0708.58024
  12. [Mr-Srz] M. Mrozek and R. Srzednicki, On time-duality of the Conley index, Results Math. 24 (1993), 161-167. Zbl0783.34039
  13. [Ryb] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin, 1987. 
  14. [Srz] R. Srzednicki, On rest points of dynamical systems, Fund. Math. 126 (1985), 69-81. Zbl0589.54049
  15. [Sz] A. Szymczak, The Conley index and symbolic dynamics, Topology 35 (1996), 287-299. Zbl0855.58023

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