Algebraic Dynamics over Global Fields: Geometric and Analytic Methods

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"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The aim of this research project is to study algebraic dynamics over global and local fields. Some of the specific goals detailed in this proposal are related to well-known open problems in the subject, including Ih's conjecture on integral preperiodic points, a function field analogue of Morton-Silverman's uniform boundedness conjecture, and the formulation of a dynamical analogue of Szpiro's conjecture for elliptic curves. The proposed techniques include the use of Berkovich analytic spaces, canonical height functions, equidistribution, classical and non-archimedean Fatou-Julia theory, moduli spaces of endomorphisms, adelic capacity theory, and Diophantine inequalities. Broadly, the majority of the ideas in this proposal can be viewed as reflections of two general themes: the first is the strong parallel between number fields and function fields of curves, and the second is the unified treatment of archimedean and non-archimedean algebraic dynamics, through the use of Berkovich analytic spaces. Algebraic dynamics is a subject which combines elements of classical dynamical systems with algebraic geometry. Its basic object of study is a set X representing the solutions to certain algebraic equations, together with a function f from X into itself. To say that these objects are defined over global and local fields means essentially that they are of special interest to number theorists and arithmetic geometers. The dynamical aspect of the subject comes into play when one repeatedly iterates the function f. For example, if x is a point of the set X for which the infinite sequence f(x), f(f(x)), f(f(f(x))), f(f(f(f(x)))), ..., eventually falls into a finite loop, then x is called a preperiodic point with respect to f. Many of the most fundamental questions in algebraic dynamics involve the nature of preperiodic points, and yet they are still not very well understood. Some examples of natural open questions which this research proposal aims to address include the following: 1. How many preperiodic points are algebraic integers? 2. Given two different functions f and g from X into itself, how many points are preperiodic with respect to both functions? 3. How rapidly can a sequence of preperiodic points approach a given non-preperiodic point?