Lyapunov Exponents, Periodic Orbits and Horseshoes for Mappings of Hilbert Spaces

Zeng, Lian; Young, Lai Sang

doi:10.1007/s00023-011-0100-9

Cited by 42 publications

(54 citation statements)

References 18 publications

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“…As an intermediate step, we extended these results to mappings of Hilbert spaces [6]. Here we go one step further, proving analogous results for semiflows on Hilbert spaces satisfying conditions consistent with those in the program outlined above.…”

Section: Summary Of Resultsmentioning

confidence: 53%

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

Zeng

Young

2012

J. Amer. Math. Soc.

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Context and motivation.This work can be seen as a small step in a program to build an ergodic theory for infinite dimensional dynamical systems, a theory the domain of applicability of which will include systems defined by evolutionary PDEs. To reduce the scope, we focus on the ergodic theory of chaotic systems, on nonuniform hyperbolic theory, to be even more specific. In finite dimensions, a basic nonuniform hyperbolic theory already exists (see e.g. [8], [9], [11], [10], [2] and [4]). This body of results taken together provides a fairly good foundation for understanding chaotic phenomena on a qualitative, theoretical level. While an infinite dimensional theory is likely to be richer and more complex, there is no reason to reinvent all material from scratch. It is thus logical to start by determining which parts of finite dimensional hyperbolic theory can be extended to infinite dimensions. Our paper is an early step (though not the first step) in this effort. With an eye toward applications to systems defined by PDEs, emphasis will be given to continuous-time systems or semiflows. Furthermore, it is natural to first consider settings compatible with dissipative parabolic PDEs, for these systems have a finite dimensional flavor (see e.g. [14], [13], and [1]). We mention some previously known results for infinite dimensional systems that form the backdrop to the present work: On the infinitesimal level, i.e. on the level of Lyapunov exponents, generalizations of Oseledets' Multiplicative Ergodic Theorem [8] to operators of Hilbert and Banach spaces have been known for some time ([12], [7], [15] and [5]). Taking nonlinearity into consideration, local results, referring to results that pertain to behavior along one orbit at a time, such as the existence of local stable and unstable manifolds, have also been proved (see e.g.[12] and [5]). This paper is among the first (see also [15]) to discuss a result of a more "global" nature. Summary of results.In this paper, we consider a specific set of results from finite dimensional hyperbolic theory and show that they can be extended (with suitable modifications) to semiflows on Hilbert spaces. The finite dimensional results in question are due to A. Katok [2]. They assert, roughly speaking, the following: Let f be a C 2 diffeomorphism of a compact Riemannian manifold, and let μ be an f -invariant Borel probability measure. Assume that (f, μ) has nonzero Lyapunov exponents and positive metric entropy. Then horseshoes are present, and that implies, among other things, an abundance of hyperbolic periodic points.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 638 ZENG LIAN AND LAI-SANG YOUNG Katok's results were proved for diffeomorphisms of compact manifolds. As an intermediate step, we extended these results to mappings of Hilbert spaces [6]. Here we go one step further, proving analogous results for semiflows on Hilbert spaces satisfying conditions consistent with those in the program outlined above. The nozero-ex...

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Section: Summary Of Resultsmentioning

confidence: 53%

“…We recall below two sets of results from [6] that apply to the discrete-time system (f, μ). 2 A. Lyapunov charts.…”

Section: Lemmamentioning

confidence: 99%

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

Zeng

Young

2012

J. Amer. Math. Soc.

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“…(See the thesis of G. Margulis [29] for compact negatively curved manifolds, and for more recent results, papers of A. Eskin-M. Mirzakhani [15] for Moduli spaces with Teichmüller flow, E. Makover -J. McGowan [26] for random manifolds, Z. Lian -L.S. Young [24] for mappings of Hilbert spaces, etc. )…”

Section: Other Topological Invariant Related To Growthmentioning

confidence: 99%

Entropy Rigidity for Metric Spaces

Lim¹

2012

The Pure and Applied Mathematics

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“…Results in this direction include the existence of Sinai-Ruelle-Bowen (SRB) measures for periodically-kicked supercritical Hopf bifurcations in a concrete PDE context [16] and the existence of horseshoes in a general context [13, 14 ].…”

Section: Introductionmentioning

confidence: 99%

Observing Lyapunov Exponents of Infinite-Dimensional Dynamical Systems

2015

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Can Lyapunov exponents of infinite-dimensional dynamical systems be observed by projecting the dynamics into ℝN using a ‘typical’ nonlinear projection map? We answer this question affirmatively by developing embedding theorems for compact invariant sets associated with C1 maps on Hilbert spaces. Examples of such discrete-time dynamical systems include time-T maps and Poincaré return maps generated by the solution semigroups of evolution partial differential equations. We make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite-dimensional dynamical system. In so doing, we adopt an empirical approach and formulate checkable conditions under which a Lyapunov exponent computed from experimental data will be a Lyapunov exponent of the infinite-dimensional dynamical system under study (provided the nonlinear projection map producing the data is typical in the sense of prevalence).

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Lyapunov Exponents, Periodic Orbits and Horseshoes for Mappings of Hilbert Spaces

Cited by 42 publications

References 18 publications

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

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

Entropy Rigidity for Metric Spaces

Observing Lyapunov Exponents of Infinite-Dimensional Dynamical Systems

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