Irregular recurrence point and chaos
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In this paper, we investigate some chaotic dynamics of a dynamical system that possesses an irregular recurrence point, i.e., a point that is Banach recurrent but not almost-periodic. We establish that a dynamical system with an irregular recurrence point is Li-Yorke chaotic for any two positive integers $r$, $s$, where $r < s$. Moreover, we demonstrate that a transitive system is thickly sensitive, and is also multi-sensitive if it admits a transitive irregular recurrence point and has at least two distinct minimal sets. Additionally, we point out that the existence of at least two distinct minimal sets is necessary.
Abstract
In this paper, we investigate some chaotic dynamics of a dynamical system that possesses an irregular recurrence point, i.e., a point that is Banach recurrent but not almost-periodic. We establish that a dynamical system with an irregular recurrence point is Li-Yorke chaotic for any two positive integers $r$, $s$, where $r < s$. Moreover, we demonstrate that a transitive system is thickly sensitive, and is also multi-sensitive if it admits a transitive irregular recurrence point and has at least two distinct minimal sets. Additionally, we point out that the existence of at least two distinct minimal sets is necessary.