# Arithmetic dynamics

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Arithmetic dynamics[1] 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, Template:Mvar-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 Template:Mvar-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

## Definitions and notation from discrete dynamics

Let Template:Mvar be a set and let F : SS be a map from Template:Mvar to itself. The iterate of Template:Mvar with itself Template:Mvar times is denoted

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

A point PS 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 Template:Mvar is the set

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

Thus Template:Mvar is preperiodic if and only if its orbit OF(P) is finite.

## Number theoretic properties of preperiodic points

Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Northcott[2] says that Template:Mvar has only finitely many Q-rational preperiodic points, i.e., Template:Mvar has only finitely many preperiodic points in P1(Q). The Uniform Boundedness Conjecture[3] of Morton and Silverman says that the number of preperiodic points of Template:Mvar in P1(Q) is bounded by a constant that depends only on the degree of Template:Mvar.

More generally, let F : PNPN be a morphism of degree at least two defined over a number field Template:Mvar. Northcott's theorem says that Template:Mvar 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 Template:Mvar, the degree of Template:Mvar, and the degree of Template:Mvar 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,[4] five,[5] or six,[6] 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.[7]

## Integer points in orbits

The orbit of a rational map may contain infinitely many integers. For example, if F(x) is a polynomial with integer coefficients and if Template:Mvar 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 Template:Mvar-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.

Theorem.[8] Let F(x) ∈ Q(x) be a rational function of degree at least two, and assume that no iterate[9] of Template:Mvar is a polynomial. Let aQ. Then the orbit OF(a) contains only finitely many integers.

## Dynamically defined points lying on subvarieties

There are general conjectures due to Shouwu 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, 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.

Conjecture. Let F : PNPN be a morphism and let CPN be an irreducible algebraic curve. Suppose that either of the following is true:
(a) Template:Mvar contains infinitely many points that are periodic points of Template:Mvar.
(b) There is a point PPN such that Template:Mvar contains infinitely many points in the orbit OF(P).
Then Template:Mvar is periodic for Template:Mvar in the sense that there is some iterate F(k) of Template:Mvar that maps Template:Mvar to itself.

## p-adic dynamics

The field of [[p-adic dynamics|Template:Mvar-adic (or nonarchimedean) dynamics]] is the study of classical dynamical questions over a field Template:Mvar that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of Template:Mvar-adic rationals Qp and the completion of its algebraic closure Cp. The metric on Template:Mvar 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,[11] which is a compact connected space that contains the totally disconnected non-locally compact field Cp.

## Generalizations

There are natural generalizations of arithmetic dynamics in which Q and Qp are replaced by number fields and their Template:Mvar-adic completions. Another natural generalization is to replace self-maps of P1 or PN with self-maps (morphisms) VV 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:

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

## Notes and references

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
2. D. G. Northcott. Periodic points on an algebraic variety. Ann. of Math. (2), 51:167--177, 1950.
3. P. Morton and J. H. Silverman. Rational periodic points of rational functions. Internat. Math. Res. Notices, (2):97--110, 1994.
4. P. Morton. Arithmetic properties of periodic points of quadratic maps. Acta Arith., 62(4):343--372, 1992.
5. E. V. Flynn, B. Poonen, and E. F. Schaefer. Cycles of quadratic polynomials and rational points on a genus-2 curve. Duke Math. J., 90(3):435--463, 1997.
6. M. Stoll, Rational 6-cycles under iteration of quadratic polynomials, 2008.
7. B. Poonen. The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Z., 228(1):11--29, 1998.
8. J. H. Silverman. Integer points, Diophantine approximation, and iteration of rational maps. Duke Math. J., 71(3):793-829, 1993.
9. An elementary theorem says that if F(x) ∈ C(x) and if some iterate of Template:Mvar is a polynomial, then already the second iterate is a polynomial.
10. S.-W. Zhang, Distributions in algebraic dynamics, Differential Geometry: A Tribute to Professor S.-S. Chern, Surv. Differ. Geom., Vol. X, Int. Press, Boston, MA, 2006, pages 381–430.
11. R. Rumely and M. Baker, Analysis and dynamics on the Berkovich projective line, ArXiv preprint, 150 pages.
12. Equidistribution in number theory, an introduction, Andrew Granville, Zeév Rudnick Springer, 2007, ISBN 978-1-4020-5403-7
13. {{#invoke:citation/CS1|citation |CitationClass=book }}