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Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert spaces

Authors: Zeng Lian and Lai-Sang Young
Journal: J. Amer. Math. Soc. 25 (2012), 637-665
MSC (2010): Primary 37DXX, 37LXX
Published electronically: March 23, 2012
MathSciNet review: 2904569
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Abstract | References | Similar Articles | Additional Information

Abstract: Two settings are considered: flows on finite dimensional Riemannian manifolds, and semiflows on Hilbert spaces with conditions consistent with those in systems defined by dissipative parabolic PDEs. Under certain assumptions on Lyapunov exponents and entropy, we prove the existence of geometric structures called horseshoes; this implies in particular the presence of infinitely many periodic solutions. For diffeomorphisms of compact manifolds, analogous results are due to A. Katok. Here we extend Katok's results to (i) continuous time and (ii) infinite dimensions.

References [Enhancements On Off] (What's this?)

  • 1. Henry, D., 1981,
    Geometric Theory of Semilinear Parabolic Equations, Springer New York. MR 610244 (83j:35084)
  • 2. Katok, A., 1980,
    Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,
    Inst. Hautes Études Sci. Publ. Math. 51 137-173. MR 573822 (81i:28022)
  • 3. Lang, S., 1993,
    Real and functional analysis,
    Springer-Verlag. MR 1216137 (94b:00005)
  • 4. Ledrappier, F. and Young, L-S., 1985,
    The metric entropy of diffeomorphisms,
    Ann. of Math. (2) 122 509-574. MR 819556 (87i:58101a)
  • 5. Lian, Z. and Lu, K., 2010,
    Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space,
    Memoirs of AMS. 206 no.967. MR 2674952 (2011g:37145)
  • 6. Lian, Z. and Young, L-S., 2011,
    Lyapunov Exponents, Periodic Orbits and Horseshoes for Mappings of Hilbert Spaces,
    Annales Henri Poincaré 12 1081-1108. MR 2823209
  • 7. Mañé, R., 1983,
    Lyapunov exponents and stable manifolds for compact transformations,
    Springer Lecture Notes in Mathematics 1007 522-577. MR 730286 (85j:58126)
  • 8. Oseledets, V. I., 1968,
    A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,
    Trans. Moscow Math. Soc. 19 197-231.
  • 9. Pesin, Y., 1977,
    Characteristic Lyapunov exponents, and smooth ergodic theory,
    Russian Math. Surveys 32 no.4 55-144. MR 0466791 (57:6667)
  • 10. Ruelle, D., 1978,
    An inequality of the entropy of differentiable maps,
    Bol. Sc. Bra. Mat. 9 83-87. MR 516310 (80f:58026)
  • 11. Ruelle, D., 1979,
    Ergodic theory of differentiable dynamical systems,
    Publ. Math., Inst. Hautes Étud. Sci. 50 27-58. MR 556581 (81f:58031)
  • 12. Ruelle, D., 1982,
    Characteristic exponents and invariant manifolds in Hilbert space,
    Ann. of Math. (2) 115 no.2 243-290. MR 647807 (83j:58097)
  • 13. Sell, G. and You, Y., 2010,
    Dynamics of evolutionary equations, Springer, New York. MR 1873467 (2003f:37001b)
  • 14. Temam, R., 1997,
    Infinite Dimensional Dynamical Systems in Mechanics and Physics,
    Applied Math. Sc., Springer-Verlag 68. MR 1441312 (98b:58056)
  • 15. Thieullen, P., 1987,
    Asymptotically compact dynamic bundles, Lyapunov exponents, entropy, dimension,
    Ann. Inst. H. Poincaré, Anal. Non Linéaire, 4 no.1 49-97. MR 877991 (88h:58064)

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Additional Information

Zeng Lian
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

Lai-Sang Young
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

Received by editor(s): February 15, 2011
Received by editor(s) in revised form: November 28, 2011
Published electronically: March 23, 2012
Additional Notes: This research was supported in part by NSF Grant DMS-0600974
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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