Arithmetic dynamics

# Arithmetic dynamics

Discussion

Encyclopedia
Arithmetic dynamics
is a field that amalgamates two areas of mathematics, dynamical systems and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.
Classically, discrete dynamics refers to the study of the iteration
Iterated function
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

of self-maps of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

or real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/or algebraic points under repeated application of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

or rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

. 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 -adic field such as Q
Arithmetic dynamics
is a field that amalgamates two areas of mathematics, dynamical systems and
number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.
Classically, discrete dynamics refers to the study of the iteration
Iterated function
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

of self-maps of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

or real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, {{math|p}}-adic, and/or algebraic points under repeated application of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

or rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

. 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 {{math|p}}-adic field such as Q
Arithmetic dynamics
is a field that amalgamates two areas of mathematics, dynamical systems and
number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.
Classically, discrete dynamics refers to the study of the iteration
Iterated function
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

of self-maps of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

or real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, {{math|p}}-adic, and/or algebraic points under repeated application of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

or rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

. 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 {{math|p}}-adic field such as Q{{math
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

or C{{math|p}} and studies chaotic behavior and the Fatou and Julia set
Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

s.

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
Periodic point
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.- Iterated functions :...

of a rational function

## Definitions and notation from discrete dynamics

Let {{math|S}} be a set and let
{{math|F}} : {{math|S}} → {{math|S}} be
a map from {{math|S}} to itself. The iterate of
{{math|F}} with itself {{math|n}} times is denoted

A point {{math|P}} ∈ {{math|S}} is periodic if
{{math|F}}({{math|n}})({{math|P}}) {{math|P}} for some {{math|n}} > 1.

The point is preperiodic if
{{math|F}}({{math|k}})({{math|P}})
is periodic for some {{math|k}} ≥ 1.

The (forward) orbit of {{math|P}} is the set

Thus {{math|P}} is preperiodic if and only if its orbit
{{math|OF}}({{math|P}}) is finite.

## Number theoretic properties of preperiodic points

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

More generally, let {{math|F}} : PN
PN be a morphism of degree at least two defined over
a number field {{math|K}}. Northcott's theorem says that
{{math|F}} has only finitely many preperiodic points in
PN({{math|K}}), and the general Uniform
Boundedness Conjecture says that the number of preperiodic points in
P{{math|N}}({{math|K}}) may be
bounded solely in terms of {{math|N}}, the degree of
{{math|F}}, and the degree of {{math|K}} over
Q.

The Uniform Boundedness Conjecture is not known even for quadratic
polynomials {{math|Fc}}({{math|x}}) =
{{math|x}}2+{{math|c}} over the
rational numbers Q. It is known in this case that
{{math|Fc}}({{math|x}}) 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
Bjorn Poonen
Bjorn Mikhail Poonen is a mathematician, four-time Putnam Competition winner and currently the Claude Shannon Professor of Mathematics at MIT.His research is primarily in number theory and algebraic geometry, but he has occasionally published in other subjects such as probability and computer...

has conjectured that
{{math|Fc}}({{math|x}}) 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 {{math|F}}({{math|x}}) is a
polynomial with integer coefficients and if {{math|a}} is
an integer, then it is clear that the entire orbit
{{math|O}}{{math|F}}({{math|a}})
consists of integers. Similarly, if
{{math|F}}({{math|x}}) is a rational map and
some iterate
{{math|F}}({{math|n}})({{math|x}})
is a polynomial, then every {{math|n}}th entry
in the orbit is an integer. An example of this phenomenon is the map
{{math|F}}({{math|x}}) =
1/{{math|xd}}, whose second iterate is a polynomial.
It turns out that this is the only way that an orbit can contain
infinitely many integers.

Theorem
Let {{math|F}}({{math|x}}) ∈
Q({{math|x}}) be a rational function of degree at
least two, and assume that no iterate
of {{math|F}} is a polynomial. Let
{{math|a}} ∈ Q. Then the orbit
{{math|O}}{{math|F}}({{math|a}})
contains only finitely 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
Faltings' theorem
In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by a finite extension...

, proven by Faltings
Gerd Faltings
Gerd Faltings is a German mathematician known for his work in arithmetic algebraic geometry.From 1972 to 1978, he studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathematics and in 1981 he got the venia legendi in mathematics, both from the University...

.
The following conjectures illustrate the general theory in the case that the subvariety is a curve.

Conjecture
Let {{math|F}} : PNPN be a morphism and let
{{math|C}} ⊂ PN be an irreducible algebraic curve. Suppose
that either of the following is true:

(a) {{math|C}} contains infinitely many points that are periodic points of {{math|F}}.

(b) There is a point {{math|P}} ∈ PN such that
{{math|C}} contains infinitely many points in the orbit {{math|OF}}( {{math|P}}).

Then {{math|C}} is periodic for {{math|F}} in the sense that there is some
iterate {{math|F}}({{math|k}}) of {{math|F}} that maps
{{math|C}} to itself.

The field of {{math is the study of classical dynamical questions
over a field {{math|K}} that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the
field of {{math|p}}-adic rationals Q{{math|p}} and the completion of its algebraic
closure C{{math|p}}. The metric on {{math|K}} and the standard definition of equicontinuity leads to the
usual definition of the Fatou and Julia set
Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

s of a rational map {{math|F}}({{math|x}}) ∈ {{math|K}}({{math|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,
which is a compact connected space that contains the totally disconnected non-locally compact field C{{math|p}}.

## Generalizations

There are natural generalizations of arithmetic dynamics
in which Q and Q{{math|p}} are
replaced by number fields and their {{math|p}}-adic completions.
Another natural generalization is to replace self-maps of P1 or P{{math|N}} with self-maps (morphisms)
{{math|V}} → {{math|V}}
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:
• dynamics over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s.
• dynamics over function fields
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...

such as C({{math|x}}).
• iteration of formal and {{math|p}}-adic power series.
• dynamics on Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s.
• arithmetic properties of dynamically defined moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...

s.
• equidistribution and invariant measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

• dynamics on Drinfeld modules.
• number-theoretic iteration problems that are not described by rational maps on varieties, for example, the Collatz problem.

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

• Arithmetic geometry
• Arithmetic topology
Arithmetic topology
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. In the 1960s topological interpretations of class field theory were given by John Tate based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on Étale cohomology...

• Combinatorics and dynamical systems
Combinatorics and dynamical systems
The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics. Also dynamical systems...