Invariant jets of a smooth dynamical system

Sophie Lemaire

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 3, page 379-448
  • ISSN: 0037-9484

Abstract

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The local deformations of a submanifold under the effect of a smooth dynamical system are studied with the help of Oseledets’ multiplicative ergodic theorem. Equivalence classes of submanifolds, called jets, are introduced in order to describe these local deformations. They identify submanifolds having the same approximations up to some order at a given point. For every order k , a condition on the Lyapunov exponents of the dynamical system is established insuring the convergence of the k -jet of a submanifold evolving under the action of the dynamical system. This condition can be satisfied even by stable dynamical systems. The limit is a k -jet which is invariant by the dynamical system.

How to cite

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Lemaire, Sophie. "Invariant jets of a smooth dynamical system." Bulletin de la Société Mathématique de France 129.3 (2001): 379-448. <http://eudml.org/doc/272345>.

@article{Lemaire2001,
abstract = {The local deformations of a submanifold under the effect of a smooth dynamical system are studied with the help of Oseledets’ multiplicative ergodic theorem. Equivalence classes of submanifolds, called jets, are introduced in order to describe these local deformations. They identify submanifolds having the same approximations up to some order at a given point. For every order $k$, a condition on the Lyapunov exponents of the dynamical system is established insuring the convergence of the $k$-jet of a submanifold evolving under the action of the dynamical system. This condition can be satisfied even by stable dynamical systems. The limit is a $k$-jet which is invariant by the dynamical system.},
author = {Lemaire, Sophie},
journal = {Bulletin de la Société Mathématique de France},
keywords = {random dynamical systems; Lyapunov exponents; multiplicative ergodic theory; jets; Pesin theory},
language = {eng},
number = {3},
pages = {379-448},
publisher = {Société mathématique de France},
title = {Invariant jets of a smooth dynamical system},
url = {http://eudml.org/doc/272345},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Lemaire, Sophie
TI - Invariant jets of a smooth dynamical system
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 3
SP - 379
EP - 448
AB - The local deformations of a submanifold under the effect of a smooth dynamical system are studied with the help of Oseledets’ multiplicative ergodic theorem. Equivalence classes of submanifolds, called jets, are introduced in order to describe these local deformations. They identify submanifolds having the same approximations up to some order at a given point. For every order $k$, a condition on the Lyapunov exponents of the dynamical system is established insuring the convergence of the $k$-jet of a submanifold evolving under the action of the dynamical system. This condition can be satisfied even by stable dynamical systems. The limit is a $k$-jet which is invariant by the dynamical system.
LA - eng
KW - random dynamical systems; Lyapunov exponents; multiplicative ergodic theory; jets; Pesin theory
UR - http://eudml.org/doc/272345
ER -

References

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  10. [10] F. Ledrappier – « Quelques propriétés des exposants caractéristiques », École d’été de Probabilité de Saint-Flour XII, Lect. Notes Math., vol. 1097, Springer, 1982, p. 305–396. Zbl0578.60029MR876081
  11. [11] V. Oseledets – « A multiplicative ergodic theorem. Lyapunov characteristic number for dynamical systems », Trans. Moscow Math. Soc.19 (1968), p. 197–231. Zbl0236.93034
  12. [12] Y. Pesin – « Families of invariant manifolds corresponding to nonzero characteristic exponents », Math. USSR Izvestija10 (1976), p. 1261–1305. Zbl0383.58012
  13. [13] —, « Characteristic Lyapunov exponents and smooth ergodic theory », Russian Math. Surveys32 (1977), p. 55–114. Zbl0383.58011
  14. [14] D. Ruelle – « Ergodic theory of differentiable dynamical systems », Publ. Math. I.H.E.S.50 (1979), p. 27–58. Zbl0426.58014MR556581

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