Reiterative distributional chaos on Banach spaces

If we change the upper and lower density in the definition of distributional chaos of a continuous linear operator on Banach space by the Banach upper and Banach lower density, respectively, we obtain Li-Yorke chaos. Motivated by this fact, we introduce the notions of reiterative distributional chaos of types $1$, $1^+$ and $2$ for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and give conditions for the existence of a vector subspace $Y$ of $X$ such that every non-zero vector in $Y$ is both irregular for $T$ and distributionally near to zero for $T.$

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