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

Lian, Zeng; Young, Lai-Sang

doi:10.1090/S0894-0347-2012-00734-6

Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert spaces
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by Zeng Lian and Lai-Sang Young

J. Amer. Math. Soc. 25 (2012), 637-665

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

Published electronically: March 23, 2012

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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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