树映射的链等价集与拓扑熵

本文讨论了树映射f的链等价集的性质,得到了f具有零拓扑熵的几个等价条件,并证明了:如果 f的一个链等价集是个无限集,那么这个链等价集的任何孤立点都是f的非周期的终于周期点.     

树映射的链等价集与拓扑熵

本文讨论了树映射f的链等价集的性质,得到了f具有零拓扑熵的几个等价条件,并证明了如果f的一个链等价集是个无限集,那么这个链等价集的任何孤立点都是f的非周期的终于周期点.     

每个点都是链回归点的树映射

设f是端点数为n的树T上的连续自映射且T上的每一点都是f的链回归点.本文证明了: (1)如果T的某个端点是f的不动点,那么,T上的每个点都是f的周期为r≤n-1的周期点,或存在自然数r ≤ n-1,使得fr含有湍流; (2)如果f的不动点都在T的内部,那么,T上的每个点都是f的周期为r≤n的周期点,或存在自然数r≤n,使得,fr含有湍流.     

区间映射的链回归点的可链点集

主要讨论区间映射的链回归点的可链点集与链等价集的关系,证明了:若区间映射的拓扑熵是零,则它的链回归点的可链点集与链等价集相等.此外还得到了区间映射有正拓扑熵的几个等价条件.     

树映射具有正拓扑熵的几个等价条件

本文讨论树映射的拓扑熵,得到树映射具有正拓扑熵的几个等价条件。     

树映射的单侧γ-极限点集与拓扑熵

本文讨论了树映射的单侧γ-极限点集与吸引中心的关系，得到了树映射具有正拓扑熵的几个等价条件．此外，还得到了树映射是强非混沌以及逐片单调树映射的拓扑熵为零的几个等价条件．     

树映射的非游荡集与拓扑混合性

设 T是个树 ,C0 ( T)表示 T上所有的连续自映射 (即 :树映射 )的集合 ,W={ fn:n≥ 2是自然数 ,f∈ C0 ( T) } .讨论了每一点都是非游荡点的树映射的性质 ,并证明了 :若混合映射 f∈ W( W在 C0 ( T)内的闭包 )且 T的每个端点都不是 f的不动点 ,则存在 g∈ C0 ( T)及自然数 k>1使 f=gk.     

树映射的不稳定流形,非游荡集与拓扑熵

设f是个端点数为n的树T上的连续自映射.本文得到了f的单侧不稳定流形与拓扑熵的关系,并证明了:（1）如果x∈i=0∞fi（Ω（f））-P（f）,那么,x的轨道是无限的;（2）如果f有一组可循环的不动点,那么h（f）≥In2（n-1）.     

树映射有异状点的一个充要条件

讨论了树上连续自映射的拓扑熵与非稳定流形之间的关系． 证明了：树上连续自映射有异状点的充要条件是其拓扑熵大于零． 因而推广了区间上连续自映射的一个结果．     

A zero topological entropy map with recurrent points not

We show that there is a continuous map of the unit interval into itself of type which has a trajectory disjoint from the set of recurrent points of , but contained in the closure of . In particular, is not closed. A function of type , with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361-366]. However, there is no trajectory contained in , since any point in is eventually mapped into . Moreover, our construction is simpler.

We use to show that there is a continuous map of the interval of type for which the set of recurrent points is not an set. This example disproves a conjecture of A. N. Sharkovsky et al., from 1989. We also provide another application of .

     

On the measure of scrambled sets of tree maps with zero topological entropy

In this paper, we show that if f is a tree map of zero topological entropy and μ is an f-invariant Borel measure then any scrambled set S has zero outer μ-measure (hence μ-measurable). In particular, if S is measurable, it has zero μ-measure.     

Topological entropy of a graph map

Let G be a graph and f : G → G be a continuous map. Denote by h(f), P(f), AP(f), R(f)and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper,we show that the following statements are equivalent:(1) h(f) 0.(2) There exists an x ∈ G such that ω(x, f) ∩ P(f) = ? and ω(x, f) is an infinite set.(3) There exists an x ∈ G such that ω(x, f)contains two minimal sets.(4) There exist x, y ∈ G such that ω(x, f)-ω(y, f) is an uncountable set and ω(y, f) ∩ω(x, f) = ?.(5) There exist an x ∈ G and a closed subset A ? ω(x, f) with f(A) ? A such that ω(x, f)-A is an uncountable set.(6) R(f)-AP(f) = ?.(7) f |P(f)is not pointwise equicontinuous.     

Tree maps having chain movable fixed points

In this paper we discuss some basic properties of chain reachable sets and chain equivalent sets of continuous maps. It is proved that if f:T→T is a tree map which has a chain movable fixed point v, and the chain equivalent set CE(v,f) is not contained in the set P(f) of periodic points of f, then there exists a positive integer p not greater than the number of points in the set End([CE(v,f)])−Pv(f) such that fp is turbulent, and the topological entropy . This result generalizes the corresponding results given in Block and Coven (1986) [2], Guo et al. (2003) [6], Sun and Liu (2003) [10], Ye (2000) [11], Zhang and Zeng (2004) [12]. In addition, in this paper we also consider metric spaces which may not be trees but have open subsets U such that the closures are trees. Maps of such metric spaces which have chain movable fixed points are discussed.     

Estimations of topological entropy for non-autonomous discrete systems

This paper is concerned with estimations of topological entropy for non-autonomous discrete systems. An estimation of lower bound of topological entropy for coupled-expanding systems associated with transition matrices in compact Hausdorff spaces is given. Estimations of upper and lower bounds of topological entropy for systems in compact metric spaces are obtained by their topological equi-semiconjugacy to subshifts of finite type under certain conditions. One example is provided for illustration.