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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert spaces
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by Zeng Lian and Lai-Sang Young PDF
J. Amer. Math. Soc. 25 (2012), 637-665 Request permission

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
  • Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
  • Serge Lang, Real and functional analysis, 3rd ed., Graduate Texts in Mathematics, vol. 142, Springer-Verlag, New York, 1993. MR 1216137, DOI 10.1007/978-1-4612-0897-6
  • F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, DOI 10.2307/1971328
  • Zeng Lian and Kening Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc. 206 (2010), no. 967, vi+106. MR 2674952, DOI 10.1090/S0065-9266-10-00574-0
  • Zeng Lian and Lai-Sang Young, Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces, Ann. Henri Poincaré 12 (2011), no. 6, 1081–1108. MR 2823209, DOI 10.1007/s00023-011-0100-9
  • Ricardo Mañé, Lyapounov exponents and stable manifolds for compact transformations, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 522–577. MR 730286, DOI 10.1007/BFb0061433
  • Oseledets, V. I., 1968, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 197-231.
  • Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
  • David Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), no. 1, 83–87. MR 516310, DOI 10.1007/BF02584795
  • David Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58. MR 556581
  • David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, DOI 10.2307/1971392
  • George R. Sell and Yuncheng You, Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002. MR 1873467, DOI 10.1007/978-1-4757-5037-9
  • Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3
  • P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 1, 49–97 (French, with English summary). MR 877991
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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
  • Email: lian@cims.nyu.edu
  • Lai-Sang Young
  • Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
  • Email: lsy@cims.nyu.edu
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 637-665
  • MSC (2010): Primary 37DXX, 37LXX
  • DOI: https://doi.org/10.1090/S0894-0347-2012-00734-6
  • MathSciNet review: 2904569