Integer Cantor sets and an order-two ergodic theorem
Integer Cantor sets and an order-two ergodic theorem
复制标题
整数康托集和二阶遍历定理
DOI：
10.1017/s0143385700007197
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发表时间：
1993
影响因子：
0.9
通讯作者：
Albert M. Fisher
中科院分区：
文献类型：
--
作者：
Albert M. Fisher
令表示对应于整数康托集的序列...... 101000101000000000101......在左移σ下的轨道闭包。证明了对于M上唯一的归一化非原子不变测度ρ，对任意f ∈ L1（M，ρ），对ρ-a.e. x∈ M，其中d = log 2/log 3，c是中三分之一Cantor集的右二阶密度的几乎必然值.证明使用重正化到标度流，加上将（M，σ）识别为角谷-冯诺依曼并矢里程计上的塔。
Abstract Let denote the orbit closure, under the left shift σ, of the sequence… (all zeroes)… 101000101000000000101 … corresponding to the integer Cantor set . We prove that with respect to the infinite invariant measure ρ, which is the unique normalized non-atomic invariant measure on M, for every f ∈ L1(M, ρ), for ρ-a.e. x∈ M where d = log 2/log 3, and c is the almost-sure value of the right-hand order-two density of the middle-third Cantor set. The proof uses renormalization to a scaling flow, plus identification of (M, σ) as a tower over the Kakutani-von Neumann dyadic odometer.