Mathematical Sciences: Coding and Convergence in Ergodic andTheory

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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.