Strong Forms of Sensitivity and Nonstandard Li–Yorke Chaos in Topological Dynamics

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This paper investigates strong forms of sensitivity and nonstandard Li–Yorke chaos. It first shows that cofinite [Formula: see text]-multi-sensitivity and thickly syndetic sensitivity are equivalent in three different types of systems, and addresses the inheritance of strong sensitivity in finite product systems. Criteria for the existence of [Formula: see text]-multi-[Formula: see text]-sensitivity are established, where [Formula: see text] is a vector and [Formula: see text] is a positive integer, and the two Lyapunov numbers associated with this property are shown to coincide in transitive systems. Moreover, it is shown that for dynamical systems with a specific type of sensitivity on a locally connected metric space, the corresponding [Formula: see text]-sensitivity is exhibited. Based on this result, it is proven that various sensitivity notions are equivalent on the closed interval [Formula: see text]. Regarding chaos, it is proven that a chain-mixing system with the limit shadowing property contains [Formula: see text]-Li–Yorke pairs for any [Formula: see text]. Furthermore, it is shown that the full shift system exhibits [Formula: see text]-Li–Yorke chaos for any [Formula: see text], and [Formula: see text]-Li–Yorke chaos is conjugacy-invariant. Finally, the relationship between standard and nonstandard Li–Yorke scrambled sets is explored through examples.

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