Journal of the
Korean Mathematical Society JKMS

In this paper, we introduce the concepts of (quasi-)weakly almost periodic point and minimal center of attraction for $\mathbb{Z}^d_+$-actions, explore the connections of levels of the topological structure the orbits of (quasi-)weakly almost periodic points and discuss the relations between (quasi-)weakly almost periodic point and minimal center of attraction. Especially, we investigate the chaotic dynamics near or inside the minimal center of attraction of a point in the cases of $S$-generic setting and non $S$-generic setting, respectively. Actually, we show that weakly almost periodic points and quasi-weakly almost periodic points have distinct topological structures of the orbits and we prove that if the minimal center of attraction of a point is non $S$-generic, then there exist certain Li-Yorke chaotic properties inside the involved minimal center of attraction and sensitivity near the involved minimal center of attraction; if the minimal center of attraction of a point is $S$-generic, then there exist stronger Li-Yorke chaotic (Auslander-Yorke chaotic) dynamics and sensitivity ($\aleph_0$-sensitivity) in the involved minimal center of attraction.

Keywords: Weakly almost periodic point, quasi-weakly almost periodic point, minimal center of attraction, chaotic dynamics

MSC numbers: Primary 58B34, 58J42, 81T75

Supported by: This work was financially supported by the National Natural Science Foundation of China (No. 12061043, 11661054).