Mathematical Sciences: Recurrence, Convergence and Entropy in Ergodic Theory

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9401093 Bergelson The purposes of this project are to 1) to study types of multiple recurrence for dynamical systems, including various aspects and refinements of the polynomial Szemeredi theorem of Bergelson and Liebman, by combinatorial and structural methods, 2) to examine the existence of bounds for various measures of the stability of averages in ergodic theory, including particular square functions for differences of various processes, through measure-theoretic and Fourier transform techniques, and also 3) to consider problems in entropy, isomorphism, and multiple mixing for different group actions in measure spaces, using both the methods introduced for studying entropy in dynamical systems with group actions, and the appropriate number-theoretic estimates that apply to such systems. The purpose of this project is to understand better the long-term behavior of dynamical systems from a variety of viewpoints. First, by examining the nature of the underlying structure of the dynamical system, types of many-fold repetition of recurrent behavior will be studied. Second, using harmonic analysis methods, the stability and rate of convergence of long term averages in the dynamical system will be considered. Third, by methods of integration and number theory, the nature and extent of chaotic behavior of dynamical systems with many modes of transformation will be better understood. ***