Mathematical Sciences: Coding and Convergence in Ergodic andTheory

彼得森将继续他对不寻常的逐点遍历定理的研究，在这些定理中，采样沿着可能不是整数的时间序列进行，像在希尔伯特变换中那样应用奇异权重，沿子序列形成平均值，或者像在Wiener-Wintner定理中那样允许复数乘数。我们将探讨这些问题与酉流的谱性质和调和分析之间的联系。由于保测变换的谱测度的光滑性尚待确定，因此提出了一种统一的策略，可以一次得到许多这样的定理。彼得森还将继续他在符号动力学方面的工作，特别是对于可数字母表和磁记录中出现的系统。这个项目是遍历理论中的数学研究。这一理论关注的是，当一个合适的底层空间的适当转换被多次迭代时，长期平均会发生什么。(通常，底层空间可能只是一个线段，转换可能会将线段分割成多个片段，然后拉伸或收缩并重新排列它们。)一种观点是研究函数在空间上的行为，当它们的自变量反复经历变换时。经常会发生函数迭代的平均值以有意义的方式收敛的情况。当可以从流中获得连续迭代次数时，在一定情况下，平均迭代次数的极限是该流上的连续平均值。Petersen感兴趣的是，除了给出原始迭代序列的整数次之外，在其他时间对流进行采样时会发生什么。

Petersen will continue his investigation of unusual pointwise ergodic theorems, in which sampling is done along a sequence of possibly non-integral times, singular weights are applied as in Hilbert transforms, averages are formed along a subsequence, or complex multipliers are allowed, as in Wiener- Wintner theorems. The connections of these questions with spectral properties of unitary flows and harmonic analysis will be explored. A unifying strategy is proposed for obtaining many such theorems at once as a consequence of a still to be determined smoothness property of spectral measures of measure- preserving transformations. Petersen will also continue his work on symbolic dynamics, especially for countable alphabets and for systems that arise in magnetic recording. This project is mathematical research in ergodic theory. This theory is concerned with what happens on average over the long run when a suitable transformation of a suitable underlying space is iterated many times. (Typically, the underlying space might be just a line segment, and the transformation might cut the line segment up into pieces, then stretch or shrink and rearrange them.) One point of view is to study the behavior of functions on the space when their arguments are repeatedly subjected to the transformation. It frequently happens that the averages of the iterates of the functions converge in a meaningful fashion. When the succesive iterates can be gotten from a flow, with continuous instead of discrete time, the limit of the averaged iterates is under certain circumstances the continuous average over the flow. Petersen is interested in what happens when the flow is sampled at other times besides the integer times that give the original sequence of iterates.