论文标题
Furstenberg耐力场序列和应用的系统
Furstenberg systems of Hardy field sequences and applications
论文作者
论文摘要
我们研究测量保存系统，称为furstenberg系统，该系统模拟了由平滑函数定义的序列的统计行为，其最多是多项式生长。典型示例是序列$（n^\ frac {3} {2}）$，$（n \ log {n}）$，和$（[n^\ frac {3} {2} {2} {2}]α）$，$α\ in \ mathb {r} $ \ mod {1} $。我们表明，它们的furstenberg系统是由有限维托里（Tori）上的一能转换而产生的，具有一些不变的措施，这些措施与HAAR度量绝对是连续的，并推断出它们与每个Ergodic系统不相交。我们还研究了$ $ $（g（s^{n^{\ frac {3} {2}}]} y）的序列的类似问题，其中$ s $是在概率空间$（y，ν）$，$ g \ in l^\ infty（n n n y $ y $ y的概率）上的概率$（y，n n n $ g \ y symical y sympal in symical in s典型的措施。我们证明，相应的furstenberg系统是强烈的固定性，并从此推断出多个千古定理，并且是测量零熵变换的多重复发结果，而零熵的转换不满足任何通勤条件。
We study measure preserving systems, called Furstenberg systems, that model the statistical behavior of sequences defined by smooth functions with at most polynomial growth. Typical examples are the sequences $(n^\frac{3}{2})$, $(n\log{n})$, and $([n^\frac{3}{2}]α)$, $α\in \mathbb{R}\setminus\mathbb{Q}$, where the entries are taken $\mod{1}$. We show that their Furstenberg systems arise from unipotent transformations on finite dimensional tori with some invariant measure that is absolutely continuous with respect to the Haar measure and deduce that they are disjoint from every ergodic system. We also study similar problems for sequences of the form $(g(S^{[n^{\frac{3}{2}}]} y))$, where $S$ is a measure preserving transformation on the probability space $(Y,ν)$, $g\in L^\infty(ν)$, and $y$ is a typical point in $Y$. We prove that the corresponding Furstenberg systems are strongly stationary and deduce from this a multiple ergodic theorem and a multiple recurrence result for measure preserving transformations of zero entropy that do not satisfy any commutativity conditions.
