Arithmetic dynamics

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. Global arithmetic dynamics is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while local arithmetic dynamics, also called p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers C by a p-adic field such as Qp or Cp and studies chaotic behavior and the Fatou and Julia sets. The following table describes a rough correspondence between Diophantine equations, especially abelian varieties, and dynamical systems: Let S be a set and let F : S → S be a map from S to itself. The iterate of F with itself n times is denoted A point P ∈ S is periodic if F(n)(P) = P for some n ≥ 1. The point is preperiodic if F(k)(P) is periodic for some k ≥ 1. The (forward) orbit of P is the set Thus P is preperiodic if and only if its orbit OF(P) is finite. Uniform boundedness conjecture for torsion points and Uniform boundedness conjecture for rational points Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Douglas Northcott says that F has only finitely many Q-rational preperiodic points, i.e., F has only finitely many preperiodic points in P1(Q). The uniform boundedness conjecture for preperiodic points of Patrick Morton and Joseph Silverman says that the number of preperiodic points of F in P1(Q) is bounded by a constant that depends only on the degree of F. More generally, let F : PN → PN be a morphism of degree at least two defined over a number field K.