测度空间的拓扑序列熵(英文)

Topological sequence entropy of the space of measures

给定一个拓扑动力系统(X,T),记M(X)为X上Borel概率测度的全体,其上的拓扑由弱拓扑所诱导.如果系统(X,T)具有零拓扑序列熵,则它称为拓扑-null的.对于给定的一个伪度量空间以及其上的一个自映射(不必连续) ,引入并研究沿着给定序列的拓扑熵,包括由空间上连续实值函数所诱导的伪度量.作为应用可以证明,给定一个序列A Z+,如果X为零维的,那么,系统(X,T)沿着A具有零拓扑熵当且仅当(M(X) ,T)沿着A具有零拓扑熵.特别的,当X为一个零维空间时,系统(X,T)为拓扑-null的当且仅当(M(X) ,T)为拓扑-null的.

Let （X, T） be a TDS and M（X） the space of all Borel probability measures on X equipped with the weak topology. （X, T） is topo-null if （X, T） has zero topological sequence entropy. Given a pseudo-metric space and a self-map, the topological sequence entropy was studied for a special class of pseudo-metrics induced by continuous real-valued functions on the space. As an application, it was proved that, given a sequence A lohtain in Z＋, if X is zero-dimensional then （X, T） has zero topological entropy along A if and only if （M（X）, T） has zero topological entropy along A. In particular, if X is zero dimensional then （X, T） is topo-null if and only if （M（X） , T） is topo-null.