Joint ergodicity and mixing
Joint ergodicity and mixing
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联合遍历性和混合
DOI：
10.1007/bf02792552
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
D. Berend
中科院分区：
文献类型：
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作者：
D. Berend
继续对[1]中开始的概率空间联合遍历测度保持变换的研究，并引入联合弱混合和强混合的概念。遍历变换和混合变换的各种性质被证明允许几种变换的类似物。特别强调了紧阿贝尔群的自同态的情况。主要结果是，给定这样的交换自同态 σ1σ2,...,σ, ofG，序列 ((1/N)Σn=0N−1σ1nf1·σ2nf2·····σsnfs 对于每个 f1,f2,…,fs∈L∞(G) 收敛于 L2(G)。此外，如果自同态是联合遍历的，即，如果极限上面的任何序列都是 Πi=1s∫Gf1dμ，其中 μ 是 Haar 测度，那么收敛也成立 μ-a.e。
The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ1σ2,...,σ, ofG, the sequence ((1/N)Σn=0N−1σ1nf1·σ2nf2· ··· · σsnfsconverges inL2(G) for everyf1,f2,…,fs∈L∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Πi=1s∫Gf1dμ, where μ is the Haar measure, then the convergence holds also μ-a.e.