Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds

Kruck, Amina; Reitmann, Volker

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 247-254

Abstract

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We prove a generalization of the Douady-Oesterlé theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.

How to cite

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Kruck, Amina, and Reitmann, Volker. "Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 247-254. <http://eudml.org/doc/294919>.

@inProceedings{Kruck2017,
abstract = {We prove a generalization of the Douady-Oesterlé theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.},
author = {Kruck, Amina, Reitmann, Volker},
booktitle = {Proceedings of Equadiff 14},
keywords = {Hilbert manifold, Hausdorff dimension, singular value},
location = {Bratislava},
pages = {247-254},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds},
url = {http://eudml.org/doc/294919},
year = {2017},
}

TY - CLSWK
AU - Kruck, Amina
AU - Reitmann, Volker
TI - Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 247
EP - 254
AB - We prove a generalization of the Douady-Oesterlé theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.
KW - Hilbert manifold, Hausdorff dimension, singular value
UR - http://eudml.org/doc/294919
ER -

References

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  1. Boichenko, V. A., Leonov, G. A., Reitmann, V., Dimension Theory for Ordinary Differential Equations, . Wiesbaden:Vieweg-Teubner Verlag, 2005. MR2381409
  2. Chueshov, I. D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, . ACTA. Kharkov, 1999 (in Russian). English translation: Acta, Kharkov (2002) (see http://www.emis.de/monographs/Chueshov). MR1632228
  3. Doering, C. R., Gibbon, J. D., Holm, D. D., Nicolaenko, B., Exact Lyapunov Dimension of the Universal Attractor for the Complex Ginzburg-Landau Equation, . Phys. Rev. Lett. 59, Iss. 26-28 (1987), pp. 2911–2914. 
  4. Douady, A., Oesterle, J., Dimension de Hausdorff des attracteurs, . Comptes Rendus del’Academie des Sciences Paris Serie A. 290 (1980), pp. 1135-1138. MR0585918
  5. Ghidaglia, M., Temam, R., Attractors for damped nonlinear hyperbolic equations, . J. Math. Pures Appl., 66 (1987), pp. 273–319. MR0913856
  6. Henon, M. A., A two-dimensional mapping with a strange attractor, . Commun. Math. Phys. 50 (1976), 2. pp. 69–77. MR0422932
  7. Kruck, A. V., Malykh, A. E., Reitmann, V., Upper Hausdorff dimension estimates and stratification for invariant sets of evolutionary systems on Hilbert manifolds, . Differential Equations, 2017 (to appear). MR3804278
  8. Lang, S., Differential and Riemannian Manifolds, . Springer, New York, 1995. MR1335233
  9. Leonov, G. A., Reitmann, V., Smirnova, V. B., Non-local Methods for Pendulum-like Feedback Systems, . Teubner-Texte zur Mathematik, Bd. 132, B.G. Teubner Stuttgart-Leipzig, 1992. MR1216519
  10. Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, . New York-Berlin: Springer-Verlag, 1988. MR0953967

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