Mathematical Sciences: Coding and Convergence in Ergodic Theory

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Ergodic theory is an active, central area of Modern Analysis. Its origins lie in the theoretical formulation for classical statistical mechanics at the end of the nineteenth century. The modern point of view derives from the profound mathematical theory of recurrence and ergodicity, as developed by Poincare, Birkhoff, von Neumann and other giants earlier in this century. As the subject has evolved and sophistication continually increased, ergodic theory has acquired close relationships with other branches of mathematics, such as dynamical systems, probability theory, functional analysis, number theory, differential topology, and differential geometry, and with applications as far ranging as mathematical physics, information theory, and computer design. Professor Petersen has worked on several aspects of ergodic theory, especially those connected with harmonic anlaysis, probability, and more recently information theory. A major part of his work has concerned maximal functions and almost everywhere convergence; for example, the refinement of the maximal ergodic theorem to an equality, speed of convergence in the ergodic theorem, the ergodic Hilbert transform, and the relationship with spectral theory and harmonic analysis. He has also conducted research in the applications of ergodic theory, specifically symbolic dynamics, to efficient coding of information across noisy channels. Here, Professor Petersen proposes to continue his research into both symbolic dynamics and almost everywhere convergence. In the former subject, he will consider general properties of symbolic dynamical systems on an infinite alphabet as well as a class of examples that relate to the control of signals for magnetic recording. In the latter area, his research will emphasize connections between the ergodic Hilbert transform and the spectral measure of a measure-preserving transformation.