Linearly Ordered Topological Space Is A Continuous Map Of The Generalized Periodic Points And Unstable Manifolds,

The generalized periodic points(that is, periodic point, non-wandering point, recurrent point, chain recurrent point,ω? limit point, and so on), are part of main research of dynamical system. In the real world, there are a lot of system with periodic state. the periodic orbit in the iterative process is one of the mathematical models which describing the periodical phenomenon, and it is the simplest invariant set of the map. More generally, there are non- wandering points set, recurrent points set, chain recurrent points set,ω? limit points set, etc. The respective characteristics of these generalized periodic points and the relationships between them were to some extent, reflect the essential characteristics of the dynamical system.After described the research background and significance of dynamical system, indicated that dynamical system play an important role in the research of mathematical theory of chaos, and introduced the research progress of the generalized periodic points. On these basis, this thesis first generalized the periodic point, the unstable manifold, and non-wandering point in one-dimensional dynamical system, and then analogy their properties on real line, using topological methods and techniques, obtain the following a series of results:1,Attention to periodic points, this thesis obtain a necessary and sufficient condition of 3-periodic point, and obtain the period structure of continuous self-map with finitely many periodic points. Further, the thesis extended Sharkovskii theorem on the real line to completely densely ordered linear ordered topological space(Abbreviated as CDLOTS).2,Attention to unstable manifolds, on CDLOTS, this thesis got some conclusions same with the real line. For example, the unstable manifold of a fixed point is connected; the unstable manifold of a periodic point is the union of finite intervals; the intersection of the unstable manifold of a fixed point and arbitrary neighborhood of the fixed point, by finite iteration of f , will contains the unstable manifold, In the unstable manifold of a fixed point p of a continuous self-map f with finitely many periodic points, there are no point different from p mapped to p by f , etc. The thtsis also pointed out the following result : The boundary points of the unstable manifold of any continuous self-map on CDLOTS, with the maximal element and the minimal element, are periodic points if they are not belong to the unstable manifold.3,Attention to unilateral unstable manifolds, this thesis got conclusions example as"the interval with endpoints of two adjacent fixed points of a continuous self-map is contained in the unilateral unstable manifold of one of the fixed points"and"The unstable manifold of a fixed point of a continuous self-map with finitely many periodic points, according to the order relation of the CDLOTS, is truncated to two parts by the fixed point, that is, the left unstable manifold and the right unstable manifold of the fixed point."In addition, this thesis generalized the properties of non-wandering points, got a equivalent condition of non-wandering points on general topological space. It is proved that the non-wandering set is a closed invariant set, and gave some equivalent forms of non-wandering set on a first countable Hausdorff space.Finally, the thesis summarized the work, prospected the work which is required in-depth study. Thus, the thesis laid some foundation for the research of the generalized periodic points future.