Arithmetic dynamics 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 self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, 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 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 classical 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:
|Diophantine 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|
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.
Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of 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 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, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.
The orbit of a rational map may contain infinitely many integers. For example, if F(x) is a polynomial with integer coefficients and if a is an integer, then it is clear that the entire orbit OF(a) consists of integers. Similarly, if F(x) is a rational map and some iterate F(n)(x) is a polynomial with integer coefficients, then every n-th entry in the orbit is an integer. An example of this phenomenon is the map F(x) = x−d, whose second iterate is a polynomial. It turns out that this is the only way that an orbit can contain infinitely many integers.
There are general conjectures due to Shouwu Zhang and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the Manin–Mumford conjecture, proven by Raynaud, and the Mordell–Lang conjecture, proven by Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve.
The field of p-adic (or nonarchimedean) dynamics is the study of classical dynamical questions over a field K that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of p-adic rationals Qp and the completion of its algebraic closure Cp. The metric on K and the standard definition of equicontinuity leads to the usual definition of the Fatou and Julia sets of a rational map F(x) ∈ K(x). There are many similarities between the complex and the nonarchimedean theories, but also many differences. A striking difference is that in the nonarchimedean setting, the Fatou set is always nonempty, but the Julia set may be empty. This is the reverse of what is true over the complex numbers. Nonarchimedean dynamics has been extended to Berkovich space, which is a compact connected space that contains the totally disconnected non-locally compact field Cp.
There are natural generalizations of arithmetic dynamics in which Q and Qp are replaced by number fields and their p-adic completions. Another natural generalization is to replace self-maps of P1 or PN with self-maps (morphisms) V → V of other affine or projective varieties.
There are many other problems of a number theoretic nature that appear in the setting of dynamical systems, including:
The Arithmetic Dynamics Reference List gives an extensive list of articles and books covering a wide range of arithmetical dynamical topics.