Mathematical Sciences: Construction and Convergence in Ergodic Theory

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Hawkins, Petersen, and Wierdl will continue to investigate subtle phenomena in the area of almost everywhere convergence in ergodic theory which involve coalescing uncountably many exceptional sets of measure zero into a single bad set of measure zero, namely random sampling of continuous-parameter stationary processes and ergodic theorems for randomly applied commuting maps. A characterization of situations in which the random ergodic averages always converge almost everywhere would provide a powerful extension of the Random Ergodic Theorem and would also generalize the return times theorem of Bourgain, Furstenberg, Katznelson, and Ornstein. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.