Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space

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

“…with regard to the asymptotic properties of Df n x , generalizations of Oseledets' Multiplicative Ergodic Theorem [11] to cocycles of linear maps of Hilbert and Banach spaces have been known for some time: a version of this result for compact operators of Hilbert spaces was proved in [15]; Banach space operators permitting nontrivial essential spectra were treated in [7,10,18]. These results are cited without proof in the present paper as Theorem 1 (see Sect.…”

On Spectral Characterization of Nonuniform Hyperbolicity

“…Now we are in position to state sufficient conditions for the existence of tempered exponential dichotomy. The following result was established by Lian and Lu [39]. …”

“…We emphasize that since the landmark works of Oseledets [34] and in particular Pesin [35] this theory has become one of the central themes of the modern dynamical systems theory (see [36] for a detailed exposition). We refer to [37][38][39] and references therein for the discussions regarding the various extensions of this theory to the infinite-dimensional setting.…”

“…We now recall the version of the multiplicative ergodic theorem established by Lian and Lu [39]. For the sake of simplicity, we will restrict our attention to the case of compact cocycles.…”

“…∈ Ω. Consequently, all the results of this paper are also valid in this setting. We have decided to present our results for the particular case of compact cocycles in order to focus on the novelties of the present paper and to avoid introducing background on quasicompact cocycles (which can be found in [39,41,42]). We note that the notion of a quasicompact cocycle is not hard to introduce but it is quite challenging to verify in practice that the cocycle possesses this property.…”