RUI: Heights, Dynamics, and Preperiodic Points

Amherst College, Amherst MA

Investigators

Abstract

DMS-0600878 Robert L. Benedetto This project concerns several problems from the relatively new field of number theoretic dynamics. One central question, formalized in the Uniform Boundedness Conjecture of Morton and Silverman, asks about the number of preperiodic points of a given dynamical system that happen to be rational numbers. While simple to pose, the question quickly leads to deep and subtle arithmetic problems. Over the past decade or so, there has been great progress on such questions, and a substantial amount of technical machinery, some due to the investigator, has been developed to help answer them. In particular, several recent advances in dynamics over local fields, in the capacity theory of filled Julia sets, and in the study of so-called local canonical heights and dynamical Green's functions, have opened new avenues for considering questions of arithmetic dynamics. The resulting theory has parallels both to the analytic study of complex dynamics and to the arithmetic study of rational points on elliptic curves and other algebraic varieties; indeed, it has drawn interest from specialists in both fields. Ultimately, the goal of the project is to study a type of classical Diophantine problem: the computation of the set of rational number solutions to a naturally arising set of polynomial equations. Such problems, together with the study of primes, have driven the study of number theory from Diophantus and the ancient Greeks through Fermat and into the present day. Besides the intrinsic beauty of such problems, they have since found applications in coding theory and cryptography. In addition, certain aspects arithmetic dynamics are accessible to undergraduates; the project includes REU summer research projects for undergraduates to aid in their mathematical training. Intense computer computations, especially of canonical heights and preperiodic points, are also planned, with any relevant data generated to be published or posted on a website, for the benefit of the larger research community.