RUI: Heights, Dynamics, and Preperiodic Points

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DMS-0600878Robert L. BenedettoThis project concerns several problems from the relatively new fieldof number theoretic dynamics. One central question, formalized in theUniform Boundedness Conjecture of Morton and Silverman, asks about thenumber of preperiodic points of a given dynamical system that happento be rational numbers. While simple to pose, the question quicklyleads to deep and subtle arithmetic problems. Over the past decade orso, there has been great progress on such questions, and a substantialamount of technical machinery, some due to the investigator, has beendeveloped to help answer them. In particular, several recent advancesin dynamics over local fields, in the capacity theory of filled Juliasets, and in the study of so-called local canonical heights anddynamical Green's functions, have opened new avenues for consideringquestions of arithmetic dynamics. The resulting theory has parallelsboth to the analytic study of complex dynamics and to the arithmeticstudy of rational points on elliptic curves and other algebraicvarieties; indeed, it has drawn interest from specialists in bothfields.Ultimately, the goal of the project is to study a type of classicalDiophantine problem: the computation of the set of rational numbersolutions to a naturally arising set of polynomial equations. Suchproblems, together with the study of primes, have driven the study ofnumber theory from Diophantus and the ancient Greeks through Fermatand into the present day. Besides the intrinsic beauty of suchproblems, they have since found applications in coding theory andcryptography. In addition, certain aspects arithmetic dynamics areaccessible to undergraduates; the project includes REU summer researchprojects for undergraduates to aid in their mathematical training.Intense computer computations, especially of canonical heights andpreperiodic points, are also planned, with any relevant data generatedto be published or posted on a website, for the benefit of the largerresearch community.