Unstable Manifolds of Continuous Self-Mappings

Unstable manifolds of continuous self-mappings on completely densely ordered linear ordered topological spaces (CDOLOTS) are discussed. Let be a continuous self-map. First, the interval with endpoints of two adjacent fixed points is contained in the unilateral unstable manifold of one of the endpoints. Then, by using the above conclusion, we prove that periodic points of not belong to the unstable manifold of their iteration points of (for some ), unless the iteration points are themselves. 1. Introduction The complexity of a dynamical system is a central topic of research since the introduction of the term of chaos in 1975 by Li and Yorke [1], known as Li-Yorke chaos today. As concepts relate to chaotic, generalized periodic points are intensively discussed. Throughout this paper, the sets of fixed points, periodic points, -limit points, and nonwandering points of a system are denoted by , , , and , respectively. Considering a continuous self-map on a compact interval , Block [2] proved the following results.(1-1) If has finitely many periodic points, then the period of each periodic point is a power of 2.(1-2) If is finite, then . Based on Block’s results, Xiong [3] obtained that(1-3) if and only if is closed,(1-4) if is closed, then for every point in , . In 2002, Ding and Nadler [4] showed that the invariant set of an -contractive map on a compact metric space is the same as the set of periodic points of . Furthermore, the set of periodic points of is finite and, only assuming that is locally compact, there is at most one periodic point in each component of . Forti et al. [5] gave an example of a triangular map of the unite square, , possessing periodic orbits of all periods and such that no infinite -limit set of contains a periodic point. Moreover, Forti show that there is a triangular map of type monotone on the fibres such that any recurrent point of is uniformly recurrent. And restricted to the set of its recurrent point is chaotic in the sense of Li and Yorke. Mai and Shao [6] obtained a structure theorem of graph maps without periodic points, which states that any graph map without periodic points must be topologically conjugate to one of the described class. Recently, Abbas and Rhoades [7] proved that some fixed point theorems in cone metric spaces gave the fact that in a cone with only a partial ordering, the continuous maps have no nontrivial periodic points. However, the research of generalized periodic points on topological space is very few. The current paper studies continuous self-maps on CDOLOTS. To characterize the properties of