Arithmetic dynamics^[1] is a field that amalgamates two areas of mathematics,dynamical systems andnumber theory. Part of the inspiration comes fromcomplex dynamics, the study of theiteration of self-maps of thecomplex plane or other complexalgebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties ofinteger,rational,p-adic, or algebraic points under repeated application of apolynomial orrational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.

Global arithmetic dynamics is the study of analogues of classicaldiophantine geometry in the setting of discrete dynamical systems, whilelocal arithmetic dynamics, also calledp-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbersC by ap-adic field such asQ_p orC_p and studies chaotic behavior and theFatou andJulia sets.

The following table describes a rough correspondence between Diophantine equations, especiallyabelian varieties, and dynamical systems:

LetS be a set and letF :S →S be a map fromS to itself. The iterate ofF with itselfn times is denoted

${\displaystyle F^{(n)}=F\circ F\circ \cdots \circ F.}$

A pointP ∈S isperiodic ifF⁽ⁿ⁾(P) =P for somen ≥ 1.

The point ispreperiodic ifF^(k)(P) is periodic for somek ≥ 1.

The (forward)orbit ofP is the set

${\displaystyle O_F(P)=\left\{P,F(P),F^{(2)}(P),F^{(3)}(P),\cdots \right\}.}$

ThusP is preperiodic if and only if its orbitO_F(P) is finite.

LetF(x) be a rational function of degree at least two with coefficients inQ. A theorem ofDouglas Northcott^[2] says thatF has only finitely manyQ-rational preperiodic points, i.e.,F has only finitely many preperiodic points inP¹(Q). Theuniform boundedness conjecture for preperiodic points^[3] of Patrick Morton andJoseph Silverman says that the number of preperiodic points ofF inP¹(Q) is bounded by a constant that depends only on the degree ofF.

More generally, letF :P^N →P^N be a morphism of degree at least two defined over a number fieldK. Northcott's theorem says thatF has only finitely many preperiodic points inP^N(K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points inP^N(K) may be bounded solely in terms ofN, the degree ofF, and the degree ofK overQ.

The Uniform Boundedness Conjecture is not known even for quadratic polynomialsF_c(x) =x² +c over the rational numbersQ. It is known in this case thatF_c(x) cannot have periodic points of period four,^[4] five,^[5] or six,^[6] although the result for period six is contingent on the validity of theconjecture of Birch and Swinnerton-Dyer.Bjorn Poonen has conjectured thatF_c(x) cannot have rational periodic points of any period strictly larger than three.^[7]

The orbit of a rational map may contain infinitely many integers. For example, ifF(x) is a polynomial with integer coefficients and ifa is an integer, then it is clear that the entire orbitO_F(a) consists of integers. Similarly, ifF(x) is a rational map and some iterateF⁽ⁿ⁾(x) is a polynomial with integer coefficients, then everyn-th entry in the orbit is an integer. An example of this phenomenon is the mapF(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.

Theorem.^[8] LetF(x) ∈Q(x) be a rational function of degree at least two, and assume that no iterate^[9] ofF is a polynomial. Leta ∈Q. Then the orbitO_F(a) contains only finitely many integers.

There are general conjectures due toShouwu Zhang^[10]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, theManin–Mumford conjecture, proven byMichel Raynaud, and theMordell–Lang conjecture, proven byGerd Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve.

Conjecture. LetF :P^N →P^N be a morphism and letC ⊂P^N be an irreducible algebraic curve. Suppose that there is a pointP ∈P^N such thatC contains infinitely many points in the orbitO_F(P). ThenC is periodic forF in the sense that there is some iterateF^(k) ofF that mapsC to itself.

The field ofp-adic (or nonarchimedean) dynamics is the study of classical dynamical questions over a fieldK that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field ofp-adic rationalsQ_p and the completion of its algebraic closureC_p. The metric onK and the standard definition of equicontinuity leads to the usual definition of theFatou andJulia sets of a rational mapF(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 toBerkovich space,^[11] which is a compact connected space that contains the totally disconnected non-locally compact fieldC_p.

There are natural generalizations of arithmetic dynamics in whichQ andQ_p are replaced by number fields and theirp-adic completions. Another natural generalization is to replace self-maps ofP¹ orP^N with self-maps (morphisms)V →V of other affine orprojective varieties.

There are many other problems of a number theoretic nature that appear in the setting of dynamical systems, including:

• dynamics overfinite fields.
• dynamics overfunction fields such asC(x).
• iteration of formal andp-adicpower series.
• dynamics onLie groups.
• arithmetic properties of dynamically definedmoduli spaces.
• equidistribution^[12] and invariantmeasures, especially onp-adic spaces.
• dynamics onDrinfeld modules.
• number-theoretic iteration problems that are not described by rational maps on varieties, for example, theCollatz problem.
• symbolic codings of dynamical systems based on explicit arithmetic expansions of real numbers.^[13]

TheArithmetic Dynamics Reference List gives an extensive list of articles and books covering a wide range of arithmetical dynamical topics.

• Arithmetic geometry
• Arithmetic topology
• Combinatorics and dynamical systems
• Arboreal Galois representation

• Lecture Notes on Arithmetic Dynamics Arizona Winter School, March 13–17, 2010, Joseph H. Silverman
• Chapter 15 ofA first course in dynamics: with a panorama of recent developments, Boris Hasselblatt, A. B. Katok, Cambridge University Press, 2003,ISBN 978-0-521-58750-1

• The Arithmetic of Dynamical Systems home page
• Arithmetic dynamics bibliography
• Analysis and dynamics on the Berkovich projective line
• Book review ofJoseph H. Silverman's "The Arithmetic of Dynamical Systems", reviewed byRobert L. Benedetto

• Arithmetic dynamics
• Dynamical systems
• Algebraic number theory

• Articles with short description
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