
Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of selfmaps of the complex plane or real line. Arithmetic dynamics is the study of the numbertheoretic properties of integer, rational, padic, and/or algebraic points under repeated application of a polynomial or rational function.
Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of selfmaps of the complex plane or real line. Arithmetic dynamics is the study of the numbertheoretic properties of integer, rational, padic, and/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 refers to the study of analogues of classical Diophantine geometry in the setting of discrete dynamical systems, while local arithmetic dynamics, also called padic or nonarchimedean dynamics, is an analogue of classical dynamics in which one replaces the complex numbers C by a padic 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:
D ophantine equations Dynamical systems
Rational and integer points on a variety Rational and integer points in an orbit
Points of finite order on an abelian variety Preperiodic points of a rational function.
Number theoretic properties of preperiodic points
Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Northcott[2] says that F has only finitely many Qrational preperiodic points, i.e., F has only finitely many preperiodic points in P1(Q). The Uniform Boundedness Conjecture[3] of Morton and 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. Northcott's theorem says that F has only finitely many preperiodic points in PN(K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points in PN(K) may be bounded solely in terms of N, the degree of F, and the degree of K over Q.
The Uniform Boundedness Conjecture is not known even for quadratic polynomials Fc(x) = x2+c over the rational numbers Q. It is known in this case that Fc(x) cannot have periodic points of period four, [4] five,[5] or six,[6] although the result for period six is contingent on the validity of the conjecture of Birch and SwinnertonDyer. Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.

