Convergence and Oscillation in Ergodic Theory and Harmonic Analysis

摘要9705228罗森布拉特这项建议是为了研究调和分析和遍历理论中的一些新方向。这里的主要原理是，在调和分析、遍历理论和概率理论中出现的算子类之间有基本的联系。例如，在遍历理论中，鞅的行为和平均值之间有很强的联系。对于极大函数而言，这是显而易见的，但对于两个过程的向上交叉而言，这也是一个重要的原则。利用遍历平均是适当的鞅的扰动的程度的估计，可以分析遍历和的振动性的精确估计。琼斯、考夫曼、罗森布拉特和维德尔在这方面的工作已经取得了很大的成功，但对这些性质的进一步研究还很有必要。特别地，本文提出证明遍历理论中最好的跳跃和上交叉不等式是通过将遍历平均与鞅联系起来而得到的。本着同样的精神，遍历平均的变分行为与鞅的行为密切相关。在最近的遍历理论工作中，平方函数对于度量遍历变换的一般平均的振荡的重要性已经变得非常明显。这项工作还得到了遍历和和与鞅之差的平方函数的估计。这种分析还在进行中，但对移动平均线的平方函数和相关的算子序列的分析还根本没有完成。此外，遍历理论中的差异的其他方面，如无条件，以及它们如何模仿鞅差序列的行为，都非常值得进一步研究。这笔资金用于研究动力系统和随机过程的长期行为。许多重要的物理现象，如股市的波动、超市的排队以及从远程传输源接收的数据，都表现出局部的反复无常的行为，而通过观察长期平均行为可以更好地理解这些行为。研究这种随机过程及其相互之间的联系，以及它们与其他较不随机的过程的精细结构之间的联系，是现代数学中的一个中心问题。只有通过这样的研究，我们才能更好地理解，从而能够控制或预测这种数学可以模拟的物理现象的演变。

Abstract 9705228 Rosenblatt This proposal is for research in a number of new directions in harmonic analysis and ergodic theory. The main principle at work here is that there are fundamental connections between classes of operators arising in harmonic analysis, ergodic theory, and probability theory. For example, there is a very strong connection between the behavior of martingales and the averages in ergodic theory. This is clear where maximal functions are concerned, but it also is an important principle in regards to upcrossings of both processes. Using estimates on the degree to which ergodic averages are a perturbation of suitable martingales, precise estimates on oscillation of ergodic sums can be analyzed. There has been a good deal of success in this direction in the work of Jones, Kaufman, Rosenblatt, and Wierdl, but additional studies of these properties are much needed . In particular, it is proposed here to show that the best possible jump and upcrossing inequalities in ergodic theory are the ones that can be obtained by relating the ergodic averages to martingales. In this same spirit, the variational behavior of ergodic averages is linked closely to the behavior of martingales. In recent work in ergodic theory, the importance of square functions for measuring the oscillation of general averages of ergodic transformations has become very clear. This work has also led to obtaining estimates on the square function of the difference between ergodic sums and martingales. Such analysis is yet in progress.However, the analysis of square functions of moving averages and of related operator sequences is not yet at all complete. Also, other aspects of differences in ergodic theory, like unconditionality, and how they mimic the behavior of martingale difference sequences, is well worth more investigation. This grant is to fund the study of the long term behavior of dynamical systems and stochastic processes. Many important physical phenomena, as varied as the beha vior of the stock market, lines in the supermarket, and data being received from a distant transmission source, exhibit locally erratic behavior which can be understood better by looking at the long term average behavior instead. The study of such stochastic processes and their connections with one another, as well as their connections in terms of their fine structure with other less random processes, is a central one in modern mathematics. It is only through such studies that we will be able to understand better and thus be able to control or predict the evolution of the physical phenomena which this mathematics can model.