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 refers to 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 |
Read more about Arithmetic Dynamics: Definitions and Notation From Discrete Dynamics, Number Theoretic Properties of Preperiodic Points, Integer Points in Orbits, Dynamically Defined Points Lying On Subvarieties, p-adic Dynamics, Generalizations, Other Areas in Which Number Theory and Dynamics Interact
Famous quotes containing the words arithmetic and/or dynamics:
“O! O! another stroke! that makes the third.
He stabs me to the heart against my wish.
If that be so, thy state of health is poor;
But thine arithmetic is quite correct.”
—A.E. (Alfred Edward)
“Anytime we react to behavior in our children that we dislike in ourselves, we need to proceed with extreme caution. The dynamics of everyday family life also have a way of repeating themselves.”
—Cathy Rindner Tempelsman (20th century)