Quantitative and Subsequence Ergodic Theorems

这个研究建议涉及遍历理论中的几乎处处和平均收敛问题，以及它们与调和分析和概率论的联系。第一组问题的收敛，这将涉及的平均收敛的平均值的测量上的随机过程在一个随机序列的时间是事先选定的。其他问题涉及随机产生的时间，产生沿沿着序列与大间隙采样。第二组问题涉及subsequence遍历定理的连续性来自成员的哈代字段，PI在以前的资助期间获得的结果表明，在上下文中的哈代字段，一个有意义的表征的"好“序列的测量是可能的。 事实上，由于在前一个赠款期内取得了重大进展，因此可以对良好的测量序列进行完整的定性。 这项工作的一维平均值给了我们信心，开始探索可能的高维结果，因此扩展了斯坦和魏格的工作。第三组问题是关于遍历平均的上穿越和相关的振荡行为。这一系列的研究是由主教，布尔甘，卡利科，B。韦斯等人。Wierdl和他的合作者发现了遍历理论和鞅之间的基本联系。 这一发现-它常常表现为能量平均值与某些鞅之差的有界平方函数-允许人们将鞅论的许多结果，如平方函数、大偏差或跳跃不等式，转化为遍历理论和调和分析结果。拟议的工作的一部分是扩大调查upcrossings到其他运营商，如高维奇异积分和平均在各种序列的域与眼睛上可能的扩展到组的行动。 这些研究将揭示遍历理论，调和分析和概率论之间的深刻联系。遍历理论起源于统计力学，即物质的统计描述。后者意味着，例如，不是描述一杯水中每个水分子的行为，而是满足于找到分子的平均速度，能量等。但随之而来的基本问题是：我们如何测量平均速度或能量。显然，不可能测量每个分子的速度，然后取数据的平均值。遍历定理说，选择一个分子，测量它每秒的速度就足够了，如果我们进行足够的测量并取数据的平均值，这个数字基本上就是一杯水中所有分子的平均速度。这个惊人的定理有一个缺点：它要求每秒钟都要精确地进行测量。 但在实践中，测量可能是在，比如说，1，3，4，6，11，.秒，或者更糟，在1.1，2.4，2.9，4.3，. 秒而不是1，2，3，...秒 显然，我们想知道我们是否仍然可以从测量数据中准确计算出平均速度。拟议的研究解决了关于测量的两个基本问题：1）什么更实际的时间序列（除了1，2，3，.）因为测量仍然会产生平均速度，能量等。2)一个人要做多少次测量才能得到平均速度、能量等的有效近似值？ 请注意，遍历定理只是说：“如果你做了足够的测量，你会得到关于平均速度，能量等的有用信息。”，但它并没有以任何方式说明多少是“足够”？

This proposal for research concerns questions of almost everywhere and mean convergence in ergodic theory, and their connections with harmonic analysis and probability theory. The first group of questions on convergence which will be addressed concerns the mean convergence of averages of measurements made on a stochastic process at a random sequence of times that is chosen in advance. Other questions concern randomly generated times which yield to sampling along sequences with big gaps. The second group of questions concerns subsequence ergodic theorems for subsequences coming from members of Hardy fields, Results obtained by the PI in the previous grant periods suggest that in the context of Hardy fields, a meaningful characterization of the ``good'' sequences of measurements is possible. Indeed, due to a significant advance in the previous grant period, a complete characterization of good sequences of measurements is at reach. This work for one dimensional averages gives us confidence to start exploring possible higher dimensional results, therefore extending the work of Stein and Weinger. The third group of questions concerns upcrossings and related oscillatory behavior of the ergodic averages. This line of research was initiated by Bishop, Bourgain, Kalikow, B. Weiss and others. Wierdl and his collaborators discovered a fundamental connection between Ergodic Theory and martingales. This discovery---which often manifests itself as a bounded squarefunction of the difference betweenergodic averages and certain martingales---allows one to translate many of the results of Martingale Theory, such as squarefunction, large deviation or jump inequalities, to ergodic theoretical and harmonic analytical results. Part of the proposed work is to extend the investigations on upcrossings to other operators such as higher dimensional singular integrals and averages over various sequences of domains with an eye on possible extensions to group actions. These investigations will reveal deep connections between ergodic theory, harmonic analysis and probability theory. Ergodic theory grew out of statistical mechanics, the statistical description of matter. This latter means, for example, that instead of describing the behavior of each individual water-molecule in a cup ofwater, one is satisfied with finding the average speed, energy etc. of the molecules. But then the fundamental question arises: how can we measure the average speed or energy. It is clearly impossible to measure the speed of each individual molecule and then take the mean of the data. The ergodic theorem says that it is enough to select a single molecule, measure its speed in each second, and if we make enough measurements and take the average of the data, the number will be basically the average speed of all the molecules in the cup of water. This amazing theorem has one drawback: it requires that the measurements are taken exactly at every second. But in practice, the measurements might be made at, say, 1, 3, 4, 6, 11,... seconds or, even worse, at 1.1, 2.4, 2.9, 4.3,... seconds instead of at 1, 2, 3,... seconds. Obviously, we would like to know whether we still can compute accurately the average speed from the measured data. The proposed research addresses two basic questions about measurements: 1) What more practical sequence of times (other than 1, 2, 3,...) for measurements will still yield the average speed, energy, etc.? 2) How many measurements one has to make to get a useful approximation to the average speed, energy, etc.? Note that the ergodic theorem just says "if you make enough measurements, you get useful information about average speed, energy, etc.", but it does not say in any way how many is "enough"?