Remarks on definitions of periodic points for nonautonomous dynamical system

Let $(X,f_{1,\mathrm{\infty }})$ be a nonautonomous dynamical system. In this paper we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new, very natural, definition of asymptotic periodicity. Moreover, this definition is resistant to changes of a beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with dense set of asymptotically periodic points imply sensitivity. We also show that event for unoformly convergent systems the nonautonomous analog of Sharkovsky’s Theorem is not valid for most definitions of periodic points.