Heights, Dynamics, and Decidability

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Dynamical systems are important mathematical objects of study, which are now ubiquitous within many of the physical sciences and social sciences due to their utility in modelling complex systems that vary over time.  One of the most difficult problems when dealing with dynamical systems is how to measure their complexity.  In practice, there is too much information to work with and one needs to work with simpler, easier to compute, invariants which give insight into how complex the ambient system is.  In the realm of Algebraic dynamical systems, that is, systems which are constructed from algebraic objects (varieties) and maps (morphisms) and where a map is repeatedly applied to points in the variety, it was observed by Kawaguchi and Silverman that a number theoretic notion, called the height, gives insight into the complexity of such systems. They have conjecturally made this precise by linking their arithmetic notion of complexity to a more classical invariant used within complex dynamics, called the dynamical degree.  This has gone on to become a major area of research within arithmetic dynamics within the last five years and much of this proposal builds upon their framework. This proposal deals with significantly broadening the scope of Kawaguchi and Silverman's work.  In joint work with Dragos Ghioca and Matthew Satriano, I initiated the study of dynamical sequences, which are sequences that are naturally obtained from algebraic dynamical systems.  In a sense, this construction is giving a "snapshot" of the dynamical system and these sequences encode information about the complexity of the ambient dynamical system.  In theory, having complete knowledge of all such sequences associated to a given dynamical system would be sufficient to reconstruct it entirely. This broad framework considers many classical results from number theory, algebraic combinatorics, and from the theory of differential equations. The long-term goal of my research program is to answer key questions that arise within this framework, which are motivated by fundamental problems across many different mathematical disciplines. These include questions about complexity, questions about decidability, and questions in Diophantine approximation and logic. I will answer these questions by applying recent techniques, some of which I have helped develop over the past six years, and others that have been developed by other researchers working on related problems. Answers to these questions will have important ramifications in many other mathematical disciplines, as they will in part provide an algorithmic framework for dealing with many naturally arising mathematical problems.  In addition, the proposal involves the training of undergraduate, graduate, and postdoctoral researchers, and will give trainees important mathematical expertise and both the programming and writing skills that are necessary for working in both STEM-related industries and in academia.