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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

   

 

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: 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.


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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

DOI: http://dx.doi.org/10.1090/S0894-0347-2012-00734-6
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.