Interactions between number theory and complex dynamical systems

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Dynamical systems are ubiquitous in applications of mathematics, and discrete-time systems have long been studied using the tools of complex analysis. Arithmetic dynamics is a relatively new field, seeking to bring the adelic tools of arithmetic geometry to bear on problems previously considered only from the complex holomorphic perspective, and this is the focus of our proposed research. Already we have had some success in using arithmetic geometry to provide new insight on, for example, post-critically finite dynamical systems.******In the current proposed program, we will expand our study of the arithmetic of critical orbits, relating various arithmetic invariants of critical orbits to algebro-geometric structures on the moduli space. The first major case of this is our recent proof of Silverman's Conjecture that the critical height on the moduli space of rational functions in one variable (of any degree) is commensurate to any ample Weil height on that space, connecting an easily-computed, and dynamically natural measure of complexity (the critical height) with a geometrically convenient measure which is more natural from the moduli space perspective, and allows access to the traditional tools of arithmetic geometry. Although the proof of Silverman's Conjecture is significant step forward for arithmetic dynamics, it is also only the first step in a certain direction. One should like to prove the analogue of this conjecture in higher dimensions (we have proven special cases, but the general conjecture is much further off). At the same time, there is more to do in the single-variable case. In particular, Silverman's Conjecture (or perhaps only its main corollary) is motivated by a rigidity result for post-critically finite rational functions, due to Thurston. McMullen's Theorem on stable families allows one to exhibit Thurston's rigidity theorem as one example among a class of rigidity results (in fairness, a central example), and this broader class of results suggests a natural way to generalize Silverman's Conjecture, considering not just the critical height but other related measures of post-critical complexity. At the same time, our recent proof of Silverman's Conjecture can be applied over function fields of algebraic varieties to give a refinement of McMullen's result (although not an independent proof, since McMullen's Theorem is an input). More general arithmetic results, applied over function fields, will have applications for families of maps in the complex holomorphic setting.******Finally, we will continue our development of the arithmetic dynamics of correspondences (iterating relations rather than functions), and of applications of arithmetic dynamics to the study of Drinfeld modules. These two topics tie in to the main program in multiple places, and offer a far greater range of entry points for students at all levels.