Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
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DOI：
10.2140/pjm.2011.250.1
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发表时间：
2009-12
影响因子：
0.6
通讯作者：
Tim Austin;T. Eisner;T. Tao
中科院分区：
文献类型：
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作者：
Tim Austin;T. Eisner;T. Tao
Furstenberg 递推定理（或等效的 Szemeredi 定理）可以用冯·诺依曼代数的语言表述如下：给定一个整数 k ≥ 2，一个具有自同构 α : M→M 的阿贝尔有限冯·诺依曼代数 (M,τ)，以及 M 中一个非负 a，且 τ(a) > 0，则有 liminf N→∞N−1 Σ n=1N Reτ(aαn(a)⋯α(k−1)n(a)) > 0; Host 和 Kra 后来的结果表明存在此限制。特别是，Reτ(aαn(a)⋯α(k−1)n(a)) 对于正密度集合中的所有 n 都是正的。从冯·诺依曼代数的角度来看，当阿贝尔假设被放弃时，很自然地会问这些结果还剩下什么。所有三个断言对于 k = 2 都成立，并且我们证明，当冯·诺依曼代数是渐近阿贝尔代数时，所有三个断言对于所有 k 都成立，并且当冯·诺依曼代数是遍历时，最后两个断言对于 k = 3 成立。然而，我们表明，即使具有遍历性，第一个声明也可能因 k = 3 而失败，即使假设遍历性，第二个声明也可能因 k ≥ 4 而失败，而第三个声明可能因 k = 3 而没有遍历性或 k ≥ 5 且假设遍历性为奇数而失败。第二个主张对于 k = 3 的非遍历系统仍然开放，第三个主张对于 k = 4 的遍历系统仍然开放。
The Furstenberg recurrence theorem (or equivalently Szemeredi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M,τ) with an automorphism α : M→M, and a nonnegative a in M with τ(a) > 0, one has liminf N→∞N−1 ∑ n=1N Reτ(aαn(a)⋯α(k−1)n(a)) > 0; a later result of Host and Kra shows this limit exists. In particular, Reτ(aαn(a)⋯α(k−1)n(a)) is positive for all n in a set of positive density. From the von Neumann algebra perspective, it is natural to ask to what remains of these results when the abelian hypothesis is dropped. All three claims hold for k = 2, and we show that all three claims hold for all k when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for k = 3 when the von Neumann algebra is ergodic. However, we show that the first claim can fail for k = 3 even with ergodicity, the second claim can fail for k ≥ 4 even when assuming ergodicity, and the third claim can fail for k = 3 without ergodicity, or k ≥ 5 and odd assuming ergodicity. The second claim remains open for nonergodic systems with k = 3, and the third claim remains open for ergodic systems with k = 4.