## Stable manifolds for nonuniform polynomial dichotomies.(English)Zbl 1194.47017

Let $$X$$ be a Banach space and $$(A_n)_n$$ a sequence of invertible bounded linear operators on $$X$$. For $$m\geq n$$, define
$A(m,n) := \begin{cases} A_{m-1}\cdots A_n & \text{if $$m>n$$}, \\ \text{Id} & \text{if $$m=n$$.} \end{cases}$
In this article, the authors introduce the notion of nonuniform polynomial dichotomy and ask for the existence of smooth manifolds in $$X$$ which are stable under the dynamics of a perturbation of $$\{ A(m,n) \}$$ if $$(A_n)$$ admits a nonuniform polynomial dichotomy.
More precisely, the sequence $$(A_n)$$ is said to admit a nonuniform polynomial dichotomy if there exists a sequence $$(P_n)$$ of projections on $$X$$ which commute with $$\{ A(m,n) \}$$, i.e., $$P_m A(m,n) = A(m,n) P_n$$, and there exist $$\epsilon\geq 0$$, $$b\geq 0 > a$$, $$D\geq 1$$ such that for $$m\geq n$$, we have
$\| A(m,n) P_n \| \leq D(m-n+1)^a n^\epsilon$ and $\| A(m,n)^{-1} (\text{Id} - P_m) \| \leq D (m-n+1)^{-b} m^\epsilon.$
For a sequence of perturbations $$(f,n)$$, $$f_n: X\to X$$, the authors consider the perturbed family
$F(m,n) := \begin{cases} (A_{m-1} + f_{m-1})\cdots (A_n + f_n) & \text{if $$m>n$$}, \\ \text{Id} & \text{if $$m=n$$.} \end{cases}$
For one family of perturbations, defined via growth restrictions of polynomial type, they show the existence of local Lipschitz stable manifolds. For a more restrictive family of perturbations, they establish the existence of global $$C^1$$ stable manifolds. For both families, they provide explicit examples.

### MSC:

 47A55 Perturbation theory of linear operators 37D10 Invariant manifold theory for dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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### References:

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