连续自映射及其扭扩半流的不变测度

THE INVARIANT MEASURES OF A CONTINUOUS MAP AND ITS SUSPENDED SEMI-FLOW

本文研究了紧致度量空间上连续自映射及连续半流的不变测度,并且证明了如下结论:(1)在拓扑等价的无不动点的连续半流的不变测度之间以及在连续自映射及其扭扩半流的不变测度之间存在一一对应;(2)作为(1)的应用,给出如下结论(见[2,定理2.1]):“环面上无不动点的连续流是唯一遍历的当且仅当它至多有一条周期轨”一个易接受的证明.

This paper studies the invariant measures of continuous maps and continuous semi-flows on a compact metric space. The main results are as follows: (1) there exists an one-to-one correspondence between invariant measures of mutually topologically equivalent semi-flows without fixed point, and between invariant measures of a continuous map and that of its suspended semi-flow; (2) as an application of (1), the authors give an accessible proof of the following result (see [2, Theorem 2.1]): 'a continuous flow without fixed point on the Torus is uniquely ergodic iff it has one periodic orbit at most'.