Li-Yorke 𝑛-chaos and distributional 𝑛-chaos in Banach spaces

This paper deals with the chaotic behavior of linear operators on Banach spaces in both discrete and continuous cases. The inheritances of chaos for linear operators and C 0 C_0 -semigroups are obtained. More precisely, for any positive integer n ≥ 2 n\geq 2 , both Li-Yorke n n -chaos and distributional n n -chaos are proved to be inherited under iterations for a linear operator. One further shows that a C 0 C_0 -semigroup { T t } t ≥ 0 \{T_t\}_{t\geq 0} and every single operator T t T_t share the same Li-Yorke n n -scrambled set and distributionally n n -scrambled set for any positive integer n ≥ 2 n\geq 2 . In particular, the Li-Yorke n n -chaos and distributional n n -chaos become the Li-Yorke chaos and distributional chaos when n = 2 n=2 , respectively. Some equivalent criteria for dense n n -chaos and generic n n -chaos of linear operators and C 0 C_0 -semigroups are also established for any n ≥ 2 n\geq 2 .