Stable manifolds for nonuniform polynomial dichotomies (Q1028337)

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.