CAREER: Entropy in dynamics: connections with geometry, algebraic numbers, and bioscience

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Dynamics is the branch of mathematics that studies systems that evolve under time. From its origins in the late 19th century to today, the mathematical theory of dynamical systems has been inspired by problems in adjacent areas of mathematics (e.g. number theory, geometry, probability), and other natural sciences (e.g. celestial mechanics, statistical mechanics, population biology). This project belongs to that rich tradition, and focuses on deriving fundamental results for a variety of dynamical systems arising from geometry, number theory and applied settings. The common thread that unifies these topics is an 'entropic' approach to all the problems under consideration. The entropy of a dynamical system is a fundamental invariant which measures the complexity of its orbit structure. The three main research directions of the project are: (1) to establish uniqueness of measures of maximal entropy and equilibrium states in a variety of higher dimensional settings of interest to the dynamics and wider mathematical communities; (2) to give a number-theoretic description of all the possible entropies that can be achieved within certain natural classes of dynamical systems; (3) to use information-theoretic interpretations of entropy and related quantities to give insight to problems in mathematical bioscience. The project contains a substantial program of synergistic educational activities, including the development of an after-school math enrichment program for the Ohio State University Young Scholars Program. The program will develop fundamental mathematical skills for a talented population of students from under-represented groups taken from all the major urban areas of Ohio.Part (1) of the project develops a novel approach to the theory of equilibrium states, focusing on implementation to non-uniformly and partially hyperbolic systems. The project develops novel techniques to overcome some of the traditional obstructions to developing an effective theory in these higher dimensional settings. The end goal is to use these results to derive numerous global statistical properties (central limit theorems, large deviations, etc.) for the systems within our framework. Motivating examples include geodesic flows in non-positive curvature, and the Teichmueller flow on the moduli space of quadratic differentials. Part (2) concerns the algebraic description of numbers arising as entropies of post-critically finite interval maps. A surprising relationship between the degree of the map and the algebraic properties of these numbers was identified experimentally in Thurston's final paper. A rigorous explanation of this phenomenon remains an open problem, which the PI has recast in terms of identifying zeros of certain complex functions. Part (3) investigates finitary versions of entropy, and related quantities, in the mathematical biosciences. The PI will use these quantities, interpreted as measures of complexity, as tools for detecting structure in large data sets, particularly those arising in genomic analysis and in the dynamics of biologically motivated networks.