Quantitative Ergodic Theorems; Spectra of Transfer Operators

摘要：本文的研究建议涉及遍历理论中几乎无处不在的平均收敛问题，以及它们与调和分析和概率论的联系。还考虑了矩阵系数传递算子的谱问题。Wierdl将要讨论的一些收敛问题涉及在事先选定的随机时间序列上对随机过程进行的测量的平均值的平均收敛性。其他问题涉及哈代域成员的子序列遍历定理，沿整数的基，沿具有大间隙的序列。Wierdl在之前的资助期内获得的结果表明，在Hardy fields的背景下，对“良好”测量序列的有意义的表征是可能的。Campbell和Wierdl将共同考虑另一组关于上交叉和遍历平均的相关振荡行为的收敛问题。这条研究路线是由Bishop、Bourgain、Kalikow、B. Weiss等人发起的。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——通常表现为遍历平均和某些鞅之间差异的平方函数——允许人们将鞅理论的许多结果，如平方函数、大偏差或上交叉不等式，转化为遍历理论结果。所谓的遍历定理（ET）是对物质的某种统计行为的理论描述。粗略地说，ET使得用两种完全不同的方法进行大量的物理测量成为可能，但却得到相同的结果。有一个例子是有用的：假设我们想找出一杯水中水粒子的平均速度。我们需要分别测量每个粒子的速度，然后取这些测量值的平均值。这种计算平均速度的方法被称为“空间平均”，因为我们在空间中去不同的点（粒子）进行测量。但是外星人说，如果我们选择一个粒子，并开始测量它在1秒，2秒，3秒等的速度，我们会得到相同的结果，并在足够长的时间后取这些测量的平均值。这种计算平均值的方法称为在*时间*内平均。以上是计算平均速度的两种截然不同的方法，具体选择哪种方法取决于具体情况。有两种可能性，在实验中就有很大的自由度。ET的问题是双重的：1)它没有明确规定我们应该测量单个粒子的速度多少次；外星人只是说“最终”时间平均值会接近空间平均值。2)在时间上的测量必须“精确地”在15秒、15秒、3秒等时间点进行。但在实践中，人们无法如此迅速地进行测量。如果测量是在1.1秒、2.8秒、3.2秒……我们的建议解决了上述两个问题：1)我们提出了各种方法来确定一个人需要进行多少次测量，以达到接近规定精度的空间平均值。2)我们检查了各种随机生成的时间，在这些时间进行的测量仍然会给出接近空间平均值的平均值。（注：随机生成的时间更有可能模拟实际情况。）

Abstract Wierdl This proposal for research concerns questions of almost everywhere and mean convergence in ergodic theory, and their connections with harmonic analysis and probability theory. Also considered are spectral questions for matrix-coefficient transfer operators. Some of the convergence questions which will be addressed by Wierdl concern 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 subsequence ergodic theorems for subsequences coming from members of Hardy fields, along bases for the integers, and along sequences with big gaps. Results obtained by Wierdl in the previous grant periods suggest that in the context of Hardy fields, a meaningful characterization of the ``good'' sequences of measurements is possible. Campbell and Wierdl will jointly consider another group of convergence questions concerning 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 squarefunction of the difference between ergodic averages and certain martingales---allows one to translate many of the results of Martingale Theory, such as squarefunction, large deviation or upcrossing inequalities, to ergodic theoretical results. The so called Ergodic Theorem (ET) is a theoretical description of a certain statistical behavior of matter. Loosely speaking, the ET makes it possible to make a large class of physical measurements in two entirely different ways, but arrive to the same result. It is useful to have an illustration: suppose we want to find out the average speed of the water particles in a cup of water. Well, we would have to measure the speed of each particle separately, and then take the average of these measurements. This method of calculating the average speed is called averaging in *space* since we go to different points (particles) in space to make the measurement. But the ET says, that we would arrive to the same result if we pick a *single* particle, and start measuring its speed at 1 second, at 2 second, at 3 second, etc, and after a long enough period of time take the average of these measurements. This method of calculating the average is called averaging in *time*. So the above is two very different ways of calculating average speed, and the method to be chosen depends on the circumstance. Having two possibilities gives great freedom in experiments. The problem with the ET is twofold: 1) It does not specify exactly *how many* times are we supposed to measure the speed of the single particle; the ET just says "eventually" the time average gets close to the space average. 2) The measurements in time has to be made *exactly* at 1s, 2s, 3s, etc. But in practice, one cannot make measurements so promptly. The ET says nothing if the measurements are made at, say, 1.1s, 2.8s, 3.2s, ...Our proposal addresses the above two problems: 1) we propose various ways to determine how many times does one need to make measurements in time to get close with prescribed accuracy to the space average. 2) We examine various randomly generated times at which measurements made will still give an average approaching the space average. (Note Randomly generated times are much more likely to model practical situations.)