Nonwandering Sets and Special $$\alpha $$-limit Sets of Monotone Maps on Regular Curves

Abstract

Let X be a regular curve and let $$f: X\rightarrow X$$ be a monotone map. We show that AP $$(f) = \text {R}(f) = \varOmega (f)$$ , where AP(f), $$\text {R}(f)$$ and $$\varOmega (f)$$ are the sets of almost periodic points, recurrent points and nonwandering points of f, respectively. On the other hand, we show that for every $$x\in X{\setminus } \text {P}(f)$$ , the special $$\alpha $$ -limit set $$s\alpha _{f}(x)$$ is a minimal set, where P(f) is the set of periodic points of f and that $$s\alpha _{f}(x)$$ is always closed, for every $$x\in X$$ . In addition, we prove that $$\text {SA}(f) = \text {R}(f)$$ , where $$\text {SA}(f)$$ denotes the special $$\alpha $$ -limit set of f. Further results related to the continuity of the limit maps are also obtained, we prove that the map $$\omega _{f}$$ (resp. $$\alpha _{f}$$ , resp. s $$\alpha _{f}$$ ) is continuous on $$X{\setminus } \text {P}(f)$$ (resp. $$X_{\infty }{\setminus } \text {P}(f)$$ ), where $$X_{\infty } = \underset{n\ge 0}{\cap }f^{n}(X)$$ .