On dynamical systems with the specification property
On dynamical systems with the specification property
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DOI：
10.1090/s0002-9947-1974-0352411-x
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发表时间：
1974-04
影响因子：
1.3
通讯作者：
K. Sigmund
中科院分区：
文献类型：
--
作者：
K. Sigmund
紧致度量空间X的连续变换T满足规范性质，如果可以用具有一定一致性的单周期轨道逼近不同的轨道。最近在遍历理论和统计力学中研究了许多这样的变换的例子。本文研究了不变测度与T轨道频率之间的关系。特别地，证明了每个不变测度（甚至是这些测度的闭连通子集）都有类属点，但所有类属点的集合都是X中的第一类。这推广了有关小数展开式和正规数的数论结果。
A continuous transfornation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.