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Nonuniform Input-Output Criteria for Exponential Expansiveness of Discrete Dynamical Systems and Applications

The aim of this paper is to present a new and complete study based on input-output methods for exponential expansiveness of discrete variational systems, treating both the nonuniform behavior and the uniform one. We continue the line of studies initiated in [J. Differential Equations 268 (2020), 4786-4829] and [J. Dynam. Differential Equations 34 (2022), 1107-1137], providing a new approach in exploring the expansiveness properties. We associate to a discrete variational system a control type system that consists of a family of input-output discrete-time equations defined between two sequence spaces whose norms are translations invariant and we introduce new concepts of exact admissibility that rely on nonuniform local conditions. First, by imposing some minimal hypotheses on the input space or respectively on the output space, we establish connections between the exact admissibilities and the nonuniform exponential expansiveness. Moreover, we give for the first time a complete description of the structure of exactly admissible pairs that can be considered to explore nonuniform expansive behaviors in the discrete variational case. In addition, all the hypotheses on the classes from which we choose the input or the output spaces are motivated by illustrative examples. Next, we present characterizations for uniform exponential expansiveness via certain nonuniform exact admissibilities and we generalize the recent criteria for expansiveness of dynamical systems from [J. Differential Equations 268 (2020), 4786-4829]. Furthermore, we show via relevant examples that the results are the most general and the working hypotheses cannot be weakened or removed. As application, we give general criteria for uniform exponential expansiveness of skew-product semiflows in terms of nonuniform admissibility properties with sequence spaces invariant under translations.

exponential expansiveness ; discrete variational system ; abstract sequence space ; discrete input - output system ; skew - product semiflow ; exact admissibility

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