RUI: Boundedness questions in arithmetic dynamics

Amherst College, Amherst MA

Investigators

Abstract

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)." This project concerns topics surrounding two key conjectures that have arisen in recent years in the field of number-theoretic dynamics. The first conjecture, stated by Morton and Silverman in 1994, predicts that there is some uniform upper bound for the number of preperiodic points of a given dynamical system that happen to be rational numbers. The second conjecture states that there is a uniform lower bound for the canonical height of a non-preperiodic rational point; in other words, roughly speaking, the non-preperiodicity of such a point must become obvious after a bounded number of iterations of the dynamical system. While easy to describe either by example or by formal statement, both conjectures quickly lead to deep and subtle arithmetic problems. Over the past decade or two, there has been great progress on such problems, and a substantial amount of technical machinery, some due to the investigator, has been developed to help answer them: advances in dynamics over local fields, capacity theory of Julia sets, local canonical heights, and dynamical Green's functions. 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, and it has drawn interest from specialists in both fields. At its heart, the focus of this project is a type of classical Diophantine problem: to understand the set of rational number solutions to a naturally arising set of polynomial equations. Such problems have been a major theme in the study of number theory for thousands of years. More specifically, a central goal of this project is to further our theoretical understanding of Diophantine problems arising from a certain kind of dynamics: the iteration of a polynomial or rational function. At the same time, because this dynamical subfield of number theory is relatively young, many of its aspects are accessible to undergraduates but still remain unsolved problems. Thus, as in an earlier successful project, the investigator plans to supervise some students in an REU summer research project to aid in their mathematical training. The REU will include intense computer computations, especially of canonical heights and preperiodic points, and any relevant data generated will be disseminated to the research community via publications and the web.