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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'' 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 by a -adic field such as 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 (forward) ''orbit of'' is the set : Thus is preperiodic if and only if its orbit is finite. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「arithmetic dynamics」の詳細全文を読む スポンサード リンク
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