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Robustness of nonuniform behavior for discrete dynamics. (English) Zbl 1291.37050

The author establishes the robustness of so-called nonuniform \((\mu,\nu)\) contractions and \((\mu,\nu)\) dichotomies for nonautonomous discrete dynamical system \({\mathcal A}(m,n)\) obtained from the product of a sequence of bounded invertible linear operators \((A_m)\), \(m= 1,2,3,\dots\) on a Banach space, where the cocycle \({\mathcal A}(m,n)\) is defined for each set of positive integers \(m\), \(n\) (with \(m\geq n\)) by \[ {\mathcal A}(m,n)= \begin{cases} A_{m-1}\cdots A_n\quad &\text{if }m> n,\\ \text{Id}\quad &\text{if }m=n.\end{cases} \] Robustness means that the contractions or dichotomies persist under small perturbations.
The author’s formulation depends on prior work of Barreira and Valls who introduced the notion of nonuniform exponential dichotomies and developed a corresponding theory for continuous and discrete dynamic. The \(\mu\) an \(\nu\) functions used by the author here represent growth rates that define nonuniform contractions and dichotomies.
Readers of this paper are advised to review first the earlier papers of Barreira and Valls.

MSC:

37D99 Dynamical systems with hyperbolic behavior
93D99 Stability of control systems
37C60 Nonautonomous smooth dynamical systems
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References:

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