Sternberg theorems for random dynamical systems

Li, Weigu; Lu, Kening

doi:10.1002/cpa.20083

Cited by 53 publications

(49 citation statements)

References 42 publications

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“…Studies of non-autonomous Sternberg theorems for random systems have been undertaken in the preprint [LL04], which appeared after this paper was submitted.…”

Section: Where [T] Denotes Integer Part Of Tmentioning

confidence: 99%

Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems

Fontich

Llave

Martín

2005

Trans. Amer. Math. Soc.

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Abstract. Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to "slow manifolds", which characterize the asymptotic convergence.Let {x i } i∈N be a regular orbit of a C 2 dynamical system f . Let S be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in S are negative and that the sums of Lyapunov exponents in S do not agree with any Lyapunov exponent in the complement of S. Denote by E S x i the linear spaces spanned by the spaces associated to the Lyapunov exponents in S. We show that there are smooth manifolds W Sand. We establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non-resonance conditions. These systems of invariant manifolds are not, in general, a foliation.

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“…Studies of non-autonomous Sternberg theorems for random systems have been undertaken in the preprint [LL04], which appeared after this paper was submitted.…”

Section: Where [T] Denotes Integer Part Of Tmentioning

confidence: 99%

Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems

Fontich

Llave

Martín

2005

Trans. Amer. Math. Soc.

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“…If ϕ is a C ∞ locally tempered random diffeomorphism in R d with a hyperbolic fixed point x = 0, then ϕ is conjugate to its linear part A(ω) by the multiplicative ergodic theorem and Theorem 1.1(ii) of [10]. By the same argument, we have {0} is an isolated invariant set for ϕ and h({0}, ϕ) is the same random shift equivalence class as its linear part.…”

Section: Example 82mentioning

confidence: 82%

Conley index for random dynamical systems

Liu

2008

Journal of Differential Equations

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Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.

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“…They briefly sketch the steps of a proof, following the scheme of [8]. A detailed proof for q = ∞ was given in 2005 by W. Li and K. Lu [15]. They in fact treat hyperbolic measurable cocycles-that is, they assume only that 0 is not in the spectrum; they do not, however, extend their polynomial normal forms to the centralizer of the initial transformation.…”

Section: 1mentioning

confidence: 99%

Non-stationary smooth geometric structures for contracting measurable cocycles

Melnick

2017

Ergod. Th. Dynam. Sys.

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Abstract. We implement a differential-geometric approach to normal forms for contracting measurable cocycles to Diff q (R n , 0), q ≥ 2. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via C q changes of coordinates. These are interpreted as nonstationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain C q homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations.

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Sternberg theorems for random dynamical systems

Abstract: In this paper, we prove the smooth conjugacy theorems of Sternberg type for random dynamical systems based on their Lyapunov exponents. We also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy.

Cited by 53 publications

References 42 publications

Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems

Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems

Conley index for random dynamical systems

Non-stationary smooth geometric structures for contracting measurable cocycles

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