Combinatorial and Algebraic Structures in Dynamics

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One of the basic problems in dynamical systems, a mathematical structure that models systems such as planetary motion, is to understand what predictions can be made about the evolution of the system in the distant future, and by doing so describe the dynamical properties of the system. A natural way to study dynamical systems is via a coding mechanism, creating a simplified model for a potentially complicated system. The principal investigator will study a series of related problems aimed at understanding how various measurements of complexity of the system relate to its dynamical properties, combining dynamical methods with algebraic and combinatorial, or counting, techniques to study these questions. Along with the research goals of obtaining a deeper understanding of the connections among these fields, the principal investigator will continue work on broadening the cohort of researchers working in these areas. Efforts in this direction include organization of conferences, with numerous meetings aimed at early-career researchers in dynamics, continuation of mentoring programs aimed at diversifying the workforce, and directing undergraduate and graduate students in research. The research carried out consists of the principal investigator building on past results to study topological and ergodic properties of symbolic systems. One series of questions is on their automorphism groups, a way to capture the symmetries of the system, exploring the complicated group structure that arises for more complicated systems and the constraints on the group structure that arise in simpler systems. A second area of focus is on the measurable properties of symbolic systems, studying how the simplex of invariant measures depends on various notions of complexity. The third focus is on higher dimensional systems, relating the complexity of configurations to the dynamical properties of the systems, and the fourth area is the study of nilpotent structures in recurrence and convergence problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.