Mathematical Sciences: Coding and Convergence in Ergodic andTheory

彼得森将继续调查 逐点遍历定理，其中采样是沿着沿着a 序列的可能非整数倍，奇异权重是 如在希尔伯特变换中那样应用，沿着沿着a 子序列，或复杂的乘数是允许的，如在维纳- Wintner定理这些问题与 酉流和调和分析的谱性质将 被探索。提出了一种统一的战略，以获得许多 这样的定理立即作为一个仍然是 测度的谱测度的确定光滑性 保持变换。彼得森还将继续他的工作 关于符号动力学，特别是可数字母和 在磁记录中出现的系统。 本课题是遍历理论的数学研究。 这一理论关注的是， 从长远来看，当一个合适的底层 空间被多次迭代。（通常，底层空间 可能只是一条线段， 将线段分割成多个部分，然后拉伸或收缩， 重新排列它们）。一种观点是研究 函数在空间上的参数重复时， 受到改造。经常发生的是， 函数迭代的平均值收敛于a 有意义的时尚当递归迭代可以得到 从一个连续而不是离散时间的流动中， 在某些情况下， 在流量上连续平均。彼得森感兴趣的是 当在除此之外的其他时间对流量进行采样时，会发生这种情况 整数倍，给出原始的迭代序列。

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.