遍历测度的一个定理及其应用

A Theorem on Ergodic Measures and Its Applications

设f是紧度量空间上的连续自映射。本文证明,如果f的所有非渐近周期的非游荡点的集合的基数是可列的,则f的遍历测度是它的周期轨道原子测度,且f的拓扑熵为零。作为推论还得到,逐点周期映射有零拓扑熵。另外,当f没有周期点时,其非游荡点的集合的基数是不可列的。

Let f be a continuous self-map on a compact metric space. We prove that if the cardinality of the set of all nonwandering points of f which are not asympto- tically periodic is countable, then its every ergodic measure is its periodic orbit atomic measure and its topological entropy vanishes. As a consequence, we get that, each pointwise periodic map has zero topological entropy. We also prove that if f has no peirodic point, then its nonwandering set is uncountable.