Strict Ergodicity and Transformation of the Torus
Strict Ergodicity and Transformation of the Torus
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DOI：
10.2307/2372899
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发表时间：
1961-10
影响因子：
1.7
通讯作者：
H. Furstenberg
中科院分区：
文献类型：
--
作者：
H. Furstenberg
引言。如果\(T\)是一个具有测度\(\mu\)的概率空间\(\Omega\)的保测变换，遍历定理保证了对于可积函数\(f\)，平均\(\lim_{N\to\infty}\frac{1}{N}\sum_{n = 0}^{N - 1}f(T^n\omega)\)关于\(\mu\)几乎处处存在。如果\(\Omega\)是一个紧拓扑空间，\(T\)是\(\Omega\)自身的一个合适的同胚，并且\(f\)是\(\Omega\)上的一个连续函数，这个陈述能否被改进呢？特别地，几乎处处收敛能否被处处收敛所替代呢？在接下来的内容中，我们将针对\(\Omega\)是一个\(r\)维环面的情况来研究这个问题。当\(r = 1\)，即\(\Omega\)是一个圆时，答案是肯定的，并且所讨论的平均总是存在（这一事实在丹若伊[1]和范坎彭[5]的结果中是隐含的）。然而，对于\(r>1\)，必须对变换\(T\)施加进一步的限制，并且我们的部分目标将是展示一类使得这种强化形式的遍历定理成立的\(T\)。我们正在考虑的问题与一个变换的“严格遍历性”问题密切相关。一个紧豪斯多夫空间\(\Omega\)的变换\(T\)是严格遍历的，如果它使\(\Omega\)的博雷尔域上的一个唯一的概率测度保持不变。这个概念首先是在克里洛夫和博戈柳博夫的动力系统理论中被引入的（[6]；也可参见[8]，[9]）。当\(T\)是一个严格遍历变换时，那么（定理1.1）对于连续的\(f\)和所有\(\omega\in\Omega\)，\(\lim_{N\to\infty}\frac{1}{N}\sum_{n = 0}^{N - 1}f(T^n\omega)\)的极限必然存在，而且，这个极限与\(\omega\)无关。在严格遍历变换的情况下，这些结论实际上比通常的遍历定理要基础得多。因此，询问一个给定空间的变换何时是严格遍历的是很自然的。正如我们将会看到的，我们一些结论成立的一个重要条件是变换\(T\)与恒等变换不同伦。这意味着变换\(T\)不能被嵌入到一个连续变换群\(T(t)\)中，所以，特别地，它不可能由环面上动力系统的考虑而产生。同伦
Introduction. If T is a measure preserving transformation ofl a probability space Q with measure Iu, the ergodic theorem assures the existence N-1 almost everywhere with respect to /i of the average limN-'Ef(T i ), where Noo n=O f is an integrable function. Can this statement be improved in case Q is a compact topological space, T a suitable homeomorphism of ?2 with itself, and f a continuous function on Q? In particular, can convergence almost everywhere be replaced by convergence everywhere? In what follows we shall examine this question for the case that Q is an r-dimensional torus. When r = 1, i. e. Q is a circle, the answer is in the affirmative and the averages in question always exist (a fact implicit in the results of Denjoy [1] and van Kampen [5]). For r > 1, however, further restrictions must be imposed on the transformation T and part of our objective will be to exhibit a class of T for which this sharpened form of the ergodic theorem holds. The question we are considering is closely tied up with that of the "strict ergodicity" of a transformation. A transformation T of a compact Hausdorff space Q is strictly ergodic if it leaves invariant a unique probability measure on the borel field of ?. This notion was first introduced in conlnection with the theory of dynamical systems by Kryloff and Bogoliuboff ([6]; cf. also [8], [9]). When T is a strictly ergodic transformation, then (Theorem N-1 1. 1) the limits of N-1 E f(TAw) necessarily exist for f continuous and all n=O X C EQ, and moreover, this limit is independent of w. In the case of a strictly ergodic transformation, these conclusions are in fact a good deal more elementary than the usual ergodic theorem. Thus it is quite natural to inquire when a transformation of a given space will be strictly ergodic. As we will see, an important condition for the validity of some of our conclusions is that the transformation T not be homotopic to the identity transformation. This implies that the transformation T cannot be embedded in a continuous transformation group T(t) and so, in particular, could :not arise from the consideration of dynamical systems on the torus. The homotopy