The fundamental theorem of dynamical systems

Douglas E. Norton

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 3, page 585-597
  • ISSN: 0010-2628

Abstract

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We propose the title of The Fundamental Theorem of Dynamical Systems for a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems are defined. First, we briefly set the context and state the theorem. After some definitions and preliminary results, based both on Conley's work and modifications to it, we present a sketch of a proof of the result in the setting of the iteration of continuous functions on compact metric spaces. Finally, we claim that this theorem should be called The Fundamental Theorem of Dynamical Systems.

How to cite

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Norton, Douglas E.. "The fundamental theorem of dynamical systems." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 585-597. <http://eudml.org/doc/247765>.

@article{Norton1995,
abstract = {We propose the title of The Fundamental Theorem of Dynamical Systems for a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems are defined. First, we briefly set the context and state the theorem. After some definitions and preliminary results, based both on Conley's work and modifications to it, we present a sketch of a proof of the result in the setting of the iteration of continuous functions on compact metric spaces. Finally, we claim that this theorem should be called The Fundamental Theorem of Dynamical Systems.},
author = {Norton, Douglas E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {chain recurrent set; attractor; decomposition; decomposition of a flow; chain recurrent set; attractor; Conley theorem; dynamical systems},
language = {eng},
number = {3},
pages = {585-597},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The fundamental theorem of dynamical systems},
url = {http://eudml.org/doc/247765},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Norton, Douglas E.
TI - The fundamental theorem of dynamical systems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 3
SP - 585
EP - 597
AB - We propose the title of The Fundamental Theorem of Dynamical Systems for a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems are defined. First, we briefly set the context and state the theorem. After some definitions and preliminary results, based both on Conley's work and modifications to it, we present a sketch of a proof of the result in the setting of the iteration of continuous functions on compact metric spaces. Finally, we claim that this theorem should be called The Fundamental Theorem of Dynamical Systems.
LA - eng
KW - chain recurrent set; attractor; decomposition; decomposition of a flow; chain recurrent set; attractor; Conley theorem; dynamical systems
UR - http://eudml.org/doc/247765
ER -

References

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  1. Anosov D.V., Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Proceedings of the Steklov Institute of Mathematics, Vol. 90, American Mathematical Society, Providence, R.I., 1969. Zbl0135.40402MR0242194
  2. Block L., Franke J.E., The chain recurrent set, attractors, and explosions, Ergodic Theory and Dynamical Systems 5 (1985), 321-327. (1985) Zbl0572.54037MR0805832
  3. Bowen R., Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer Verlag, New York, 1975. MR0442989
  4. Bowen R., On Axiom A Diffeomorphisms, CBMS Regional Conference Series in Mathematics, Vol. 35, American Mathematical Society, Providence, R.I., 1978. Zbl0383.58010MR0482842
  5. Conley C., The Gradient Structure of a Flow, I, IBM RC 3932, #17806, 1972; reprinted in Ergodic Theory and Dynamical Systems 8* (1988), 11-26. (1988) Zbl0687.58033MR0967626
  6. Conley C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, Vol. 38, American Mathematical Society, Providence, R.I., 1978. Zbl0397.34056MR0511133
  7. Easton R., Isolating blocks and epsilon chains for maps, Physica D 39 (1989), 95-110. (1989) Zbl0696.58042MR1021184
  8. Franks J., Book review, Ergodic Theory and Dynamical Systems 7 (1987), 313-315. (1987) MR0967632
  9. Franks J., A Variation on the Poincaré-Birkhoff Theorem, in: Hamiltonian Dynamical Systems, K.R. Meyer and D.G. Saari, eds., American Mathematical Society, Providence, R.I., 1988, pp. 111-117. Zbl0679.58026MR0986260
  10. Hurley M., Chain recurrence and attraction in non-compact spaces, Ergodic Theory and Dynamical Systems 11 (1991), 709-729. (1991) Zbl0785.58033MR1145617
  11. McGehee R.P., Some Metric Properties of Attractors with Applications to Computer Simulations of Dynamical Systems, preprint, 1988. 
  12. Milnor J., On the concept of attractor, Communications in Mathematical Physics 99 (1985), 177-195. (1985) Zbl0602.58030MR0790735
  13. Norton D.E., Coarse-Grain Dynamics and the Conley Decomposition Theorem, submitted, 1994. 
  14. Norton D.E., The Conley Decomposition Theorem for Maps: A Metric Approach, submitted, 1994. Zbl0856.58028MR1366526
  15. Norton D.E., A Metric Approach to the Conley Decomposition Theorem, Thesis, University of Minnesota, 1989. 
  16. Ruelle D., Small random perturbations of dynamical systems and the definition of attractors, Communications in Mathematical Physics 82 (1981), 137-151. (1981) Zbl0482.58017MR0638517

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