A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
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 uniform boundedness conjecture for preperiodic points[3] 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. 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.
The following conjectures illustrate the general theory in the case that the subvariety is a curve.
There are natural generalizations of arithmetic dynamics in which Q and Qp are replaced by number fields and their p-adic completions.