The Upper Capacity Topological Entropy of Free Semigroup Actions for Certain Non-compact Sets

In this paper, we first introduce some new notions of 'periodic-like' points, such as almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points, of free semigroup actions. We find that the corresponding sets and gap-sets of these points carry full upper capacity topological entropy of free semigroup actions under certain conditions. Furthermore, phi-irregular set acting on free semigroup actions is introduced and it also carries full upper capacity topological entropy in the system with specification property. Finally, we introduce the level set for local recurrence of free semigroup actions and analyze its connections with upper capacity topological entropy. Our analysis generalizes the results obtained by Tian (Different asymptotic behavior versus same dynamical complexity: recurrence & (ir)regularity. Adv. Math. 288:464-526, 2016), Chen et al. (Topological entropy for divergence points. Ergodic Theory Dynam Syst. 25:1173-1208, 2005) and Lau and Shu (The spectrum of Poincare recurrence. Ergodic Theory Dynam Syst 28:1917-1943, 2007) etc.