树映射拓扑熵为零的几个充要条件

Some sufficient and necessary conditions of the tree map with zero entropy

研究了拓扑熵为零的树 (即一维紧致连通不含圈的分支流形 )映射 ,其ω-极限集的特征 ,得到了 :设 f :T→ T是连续自映射 ,则 h(f ) =0充分且必要条件是对任意的 x∈ T,ω(x,f )或者是周期轨 ,或者是不含任何周期轨的无限集。此外 ,在系统具有伪轨追踪性质的假设下 ,得到了 h(f ) =0的另一个充分必要条件是 AP(f ) =R(f ) ,这些结果都推广了区间映射的相应结论。

The character of ω limit set of a tree map is studied when its entropy is zero.Let f:T→T be a continuous map, then h(f)=0 if and only if for any x∈T,ω(x,f) is either a periodic orbit or an infinite set with no other perodic orbits. Also, under the supposition that the tree has POTP property, the following result is obtained: h(f)=0 if only if AP(f)=R(f) .