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Nonuniform $(h,k,μ,ν)$-dichotomy and Stability of Nonautonomous Discrete Dynamics

In this paper, a new notion called the general nonuniform $(h,k,μ,ν)$-dichotomy for a sequence of linear operators is proposed, which occurs in a more natural way and is related to nonuniform hyperbolicity. Then, sufficient criteria are established for the existence of nonuniform $(h,k,μ,ν)$-dichotomy in terms of appropriate Lyapunov exponents for the sequence of linear operators. Moreover, we investigate the stability theory of sequences of non uniformly hyperbolic linear operators in Banach spaces, which admit a nonuniform $(h,k,μ,ν)$-dichotomy. In the case of linear perturbations, we investigate parameter dependence of robustness or roughness of the nonuniform $(h,k,μ,ν)$-dichotomies and show that the stable and unstable subspaces of nonuniform $(h,k,μ,ν)$-dichotomies for the linear perturbed system are Lipschitz continuous for the parameters. In the case of nonlinear perturbations, we construct a new version of the Grobman-Hartman theorem and explore the existence of parameter dependence of stable Lipschitz invariant manifolds when the nonlinear perturbation is of Lipschitz type.