Quantitative Ergodic Theorems; Spectra of Transfer Operators

抽象Wierdl 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。还考虑了 矩阵系数转移算子的谱问题。 Wierdl将讨论的一些收敛问题 关注的是在一个 随机过程，在随机时间序列中选择， 提前其他问题涉及的子序列遍历定理 来自哈代领域的成员，沿着基地的 整数和具有大间隙的沿着序列。Wierdl在以前的赠款期间获得的结果表明，在哈代领域的背景下，一个有意义的表征的“好”的测量序列是可能的。 坎贝尔和维尔德尔将共同考虑另一组趋同 有关的upcrossings和相关的振荡行为的问题， 遍历平均数这项研究是由毕晓普发起的， Bourgain，Kalikow，B.韦斯等人。 Wierdl和他的合作者 发现了遍历理论和 鞅 这一发现-经常表现为 各态历经平均值与某些 martingales-允许一个翻译的许多结果 鞅论，如平方函数，大偏差或 上交不等式，遍历理论结果。 所谓的遍历定理（ET）是对一个随机变量的理论描述。 物质的某些统计行为。 不严格地说，ET 使我们有可能进行大量的物理测量， 两种完全不同的方式，却得到了同样的结果。 有一个例子是很有用的：假设我们想知道一杯水中水粒子的平均速度。 我们必须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，因为我们要在空间中的不同点（粒子）进行测量。 但是ET说，如果我们选择一个 * 单个 * 粒子，并在1秒，2秒，3秒等开始测量它的速度，并在足够长的时间后取这些测量的平均值，我们会得到同样的结果。 这种计算平均值的方法称为在 * 时间 * 内求平均值。 所以上面是两种计算平均速度的不同方法， 选择什么样的方法取决于具体情况。 有两种可能性给实验带来了很大的自由。 ET的问题是双重的：1）它没有确切地说明我们应该做多少次 测量单个粒子的速度; ET只是说“最终”时间平均值接近空间平均值。 2)时间上的测量必须在1 s、2s、3s等处进行 * 精确 *，但在实践中，人们不能如此迅速地进行测量。 如果测量是在1.1s、2.8s、3.2s、.我们的建议针对上述两个问题： 1)我们提出了各种方法来确定一个人需要多少次 及时进行测量，以规定的精度接近 平均空间。 2)我们检查各种随机生成的时间，在这些时间进行测量仍然会给出接近空间平均值的平均值。 (Note随机生成的时间更有可能模拟实际的 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.)