Mathematical Sciences: Equivalence Relations of Measurable Dynamical Systems

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9501103 Sahin The proposed research concentrates on areas in measurable dynamics. In particular, it is proposed to study measurable dynamical systems with multidimensional time parameters. Some of the topics considered are tiling based dynamical systems and restricted orbit equivalences. Every multidimensional continuous-time dynamical system can be represented as a tiling based dynamical system. There is a natural discrete-time dynamical system associated to each such representation. This investigation will explore the intrinsic relationship between discrete dynamical systems arising from different representations of a fixed continuous-time system. Tiling based problems are interesting in their own right. They provide a link between crystallography, multidimensional continuous-time and discrete-time dynamical systems. They also make it possible to pose problems of abstract measurable dynamics as concrete tiling geometry questions. This investigation will provide insight into some of these connections. In addition, the method proposed here introduces a new one-step constructive technique. This technique differs from the traditional sequential approximation techniques used in studying higher dimensional dynamical systems. ***