Abstract:
Discrete dynamics is the study of iteration. A primary objective of dynamics is the classification of points in a set S according to their orbits under repeated application of a self-map f:S→S. Classically, S is taken to be Rⁿ or Cⁿ, and real and complex dynamics are mature and thriving fields of study. But for a number theorist, it is natural to take S to be a set of arithmetic interest such as Z or Q. The past 25 years has seen the development of the field of Arithmetic Dynamics, in which one studies dynamical analogues of classical results in number theory and arithmetic geometry. Here are two illustrative problems: Let f(z) ∈ Q(z) be a rational function. (1) There are always infinitely many complex numbers with finite forward orbit under iteration of f, but how many of these complex number can be rational numbers? (2) If we take a rational starting point α ∈ Q , when is it possible for infinitely many points in the orbit of α to be integers? In this talk I will discuss these and other problems in arithmetic dynamics. As is typical in number theory, there are many questions that are easy to state, but difficult to solve.