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, -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 by a -adic field such as P-adic number| or 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:
Definitions and notation from discrete dynamics
Let be a set and let be a map from to itself. The iterate of with itself times is denoted A point is periodic if for some. The point is preperiodic if is periodic for some. The orbit of is the set Thus is preperiodic if and only if its orbit is finite.
Let be a rational function of degree at least two with coefficients in. A theorem of Northcott says that has only finitely many -rational preperiodic points, i.e., has only finitely many preperiodic points in. The Uniform Boundedness Conjecture of Morton and Silverman says that the number of preperiodic points of in is bounded by a constant that depends only on the degree of. More generally, let be a morphism of degree at least two defined over a number field. Northcott's theorem says that has only finitely many preperiodic points in , and the general Uniform Boundedness Conjecture says that the number of preperiodic points in may be bounded solely in terms of, the degree of, and the degree of over. The Uniform Boundedness Conjecture is not known even for quadratic polynomials over the rational numbers. It is known in this case that 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 cannot have rational periodic points of any period strictly larger than three.
Integer points in orbits
The orbit of a rational map may contain infinitely many integers. For example, if is a polynomial with integer coefficients and if is an integer, then it is clear that the entire orbit consists of integers. Similarly, if is a rational map and some iterate is a polynomial with integer coefficients, then every -th entry in the orbit is an integer. An example of this phenomenon is the map, whose second iterate is a polynomial. It turns out that this is the only way that an orbit can contain infinitely many integers.
Dynamically defined points lying on subvarieties
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.
''p''-adic dynamics
The field of -adic dynamics is the study of classical dynamical questions over a field that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of -adic rationals and the completion of its algebraic closure. The metric on and the standard definition of equicontinuity leads to the usual definition of the Fatou and Julia sets of a rational map. 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.
Generalizations
There are natural generalizations of arithmetic dynamics in which and are replaced by number fields and their -adic completions. Another natural generalization is to replace self-maps of or with self-maps of other affine or projective varieties.
Other areas in which number theory and dynamics interact
There are many other problems of a number theoretic nature that appear in the setting of dynamical systems, including:
