In this paper, we introduce a new concept, (α,β)-mean Li–Yorke chaotic operator, which includes the standard mean Li–Yorke chaotic operators as special cases. We show that when α < 1 ≤ β or α ≤ 1 < β, (α,β)-mean Li–Yorke chaotic dynamics is strictly stronger than the ones that appeared in mean Li–Yorke chaos. When α > 1 or β < 1, it has completely different characteristics from the mean Li–Yorke chaos. We prove that no finite-dimensional Banach space can support (α,β)-mean Li–Yorke chaotic operators. Moreover, we show that an operator is (α,β)-mean Li–Yorke chaos if and only if there exists an (α,β)-mean semi-irregular vector for the underlying operator, and if and only if there exists an (α,β)-mean irregular vector when α ≤ 1, which generalizes the recent results by Bernardes et al. given in 2018. When α > 1, we construct a counterexample in which it is an (α,β)-mean Li–Yorke chaotic operator but does not admit an (α,β)-mean irregular vector. In addition, we show that an operator with dense generalized kernel is (α,β)-mean Li–Yorke chaotic if and only if there exists a residual set of (α,β)-mean irregular vectors, and if and only if there exists an β-mean unbounded orbit.

中文翻译：

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Banach空间上线性算子的(α,β)-Mean Li-Yorke混沌特征

在本文中，我们引入了一个新概念，(α,β)-mean Li-Yorke 混沌算子，包括标准均值 Li-Yorke 混沌算子作为特例。我们证明当α < 1 ≤ β要么α ≤ 1 < β,(α,β)-mean Li-Yorke 混沌动力学严格地强于出现在平均 Li-Yorke 混沌中的动力学。什么时候α > 1要么β < 1，它与平均的 Li-Yorke 混沌具有完全不同的特征。我们证明没有有限维 Banach 空间可以支持(α,β)-mean Li-Yorke 混沌算子。此外，我们证明了一个算子是(α,β)-mean Li-Yorke 混沌当且仅当存在(α,β)- 基础运算符的均值半不规则向量，并且当且仅当存在(α,β)-mean 不规则向量当α ≤ 1，它概括了 Bernardes 最近的结果等。在 2018 年给出。当α > 1，我们构造一个反例，其中它是(α,β)-mean Li-Yorke 混沌算子但不承认(α,β)-均值不规则向量。此外，我们证明了具有密集广义核的算子是(α,β)-mean Li-Yorke 混沌当且仅当存在残差集(α,β)-均值不规则向量，且当且仅当存在β- 平均无界轨道。