Mathematical Sciences: Ergodic Theory of Symbolic Systems

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Lind will continue his investigations of the dynamics of commuting mappings and other topics in symbolic dynamics. Lind has recently found two invariants of Markov shifts based on spanning trees. These will be further studied. Tuncel plans to continue his study of the structure and classifications of Markov chains and related positivity problems. Obstructions arising from periodic points approaching the boundary of the polytope will be studied with the aim of using the polytope and its faces in an inductive construction. 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.