Fibres dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension

Abstract

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Stability of nonautonomous differential equations in Hilbert spaces

“…respectively of the Lyapunov exponents and on H n \ {0} counted with multiplicities (see (13) and (17)). Mimicking once more the abstract theory of Lyapunov exponents in finite-dimensional spaces, we introduce the Perron coefficient of and ,…”

“…Later on Mañé [9] considered transformations in Banach spaces under some compactness assumptions (including the case of differentiable maps with compact derivative at each point). The results of Mañé were extended by Thieullen in [13] for a class of transformations satisfying a certain asymptotic compactness. In view of the regularity theory in finite-dimensional spaces one should ask, and this is another motivation for our study, whether the above "geometric" results in the infinitedimensional setting have behind them an analogous (infinite-dimensional) regularity theory, which additionally reduces to the classical theory when applied to the finitedimensional setting.…”

Observing Lyapunov Exponents of Infinite-Dimensional Dynamical Systems

“…For a smooth map f on a real Hilbert space H , Ruelle proves a multiplicative ergodic theorem for the derivative cocycle assuming Df x is compact [23]. Cocycles into operators on Banach spaces (possibly with nontrivial essential spectrum) are treated in [5, 12, 17, 26 ].…”

Stability of Hyperbolic Partial Differential Equations with Random Loads

“…On the other hand, there are extensions of the multiplicative ergodic theorem to the infinite-dimensional case, which have been proved by Ruelle [17], Marl6 [14], Thieullen [19] and Schaumlhffel [18]. The result is that even systems in blockdiagonal structure could exhibit genuinely infinite-dimensional behavior; i.e., every finitedimensional block system is stable (A ) < C < 0 for every n) while the infinitedimensional system is unstable.…”

“…A multiplicative ergodic theorem for Hilbert-space-valued cocycles. In this section we want to present a multiplicative ergodic theorem that is a generalization of a result by Whieullen [19] and has been proved by Schaumlhffel [18]. For that reason, we must introduce some definitions.…”