aa r X i v : . [ m a t h . HO ] A ug A Glimpse of Arithmetic Dynamics
Ryan Grady ∗ Mark Poston † August 7, 2019
In this note, we offer a palatable introduction to the field of arithmeticdynamics . That is, we study the patterns that arise when iterating a polynomialmap. This note is accessible to those who have taken an introductory proofbased course and some linear algebra; the appendix utilizes abstract algebra. Amore sophisticated overview of the field is given in [1].In our study, we consider polynomials defined not over fields like the realor complex numbers, but rather fields which have only finitely many elements.These finite fields arise in abstract algebra and number theory, and they are usedextensively in the mathematics underlying cryptography. We pay particularattention to the shape of the patterns arising from iterating a given polynomial,so called orbit types.We begin by introducing (discrete) dynamical systems. We offer severalexamples, discuss various types of orbits, and representations of systems bydirected graphs. Next, we recall finite fields and polynomial dynamics over finitefields. We then prove a result about when a given system has no preperiodicorbits. Finally, we describe a handful of questions for further pursuit, and anappendix where we outline the (classical) existence and construction of finitefields.
Dynamical systems are a fundamental discipline in mathematics. Moreover, dy-namical systems abound in the natural and social worlds. For instance, dynam-ics are often applied to game/equilibrium theory, which itself is used in devisingpricing schemes in economics. Chemical and biological networks/pathways areoften studied using dynamical systems as well. A standard course in dynamicalsystems is based on [6]. In what follows, we will only consider discrete dynamicalsystems, i.e., a set S and a function f : S → S .To get a feel for dynamics, consider a group of six friends: Avery, Blake,Charlie, Dakota, Emerson, and Finley. Each person has a ball and follows asimple rule for to whom to pass their ball: ∗ Montana State University † Montana State University. Mark Poston was supported during this research by a grantfrom the Undergraduate Scholars Program at Montana State. Avery always passes to Blake; • Blake always passes to Avery; • Charlie always passes to Dakota; • Dakota always passes to Emer-son; • Emerson always passes to Char-lie; • Finley always holds their ball.Pictorially the situation looks as follows.
AB C ED F
Observe that there are different types of patterns (orbits) in the picture, e.g., ittakes three rounds for the Charlie’s ball to return to them, while it only takestwo rounds for Avery’s ball to return to Avery, Finley’s ball doesn’t move at all.What we have just described is a (discrete) dynamical system , more preciselywe had a set S —in the example, let us represent the friends by their first initial,so S = { A, B, C, D, E, F } —and a function f : S → S . We then consideredthe iterates of f , i.e., the n -fold compositions of f , f n = f ◦ f ◦ · · · ◦ f . Inthe friends example, f was the rule for passing, e.g., f maps Avery to Blake, so f ( A ) = B . We then considered the forward orbits of various balls, e.g., Charlie’sball went to Dakota, then onto Emerson, then returned to Charlie. In terms ofthe function, we have f ( C ) = C , and we say that Charlie (or their ball) is a periodic point of period
3; similarly, Dakota and Emerson are periodic points ofperiod 3. In addition, Avery and Blake are both periodic points with period 2.Finley has period 1, so they are called a fixed point . We will see below anotherorbit type: a pre-periodic point , i.e., an element p of our set S which becomesperiodic after some number of iterations. 1 2 34 5 67 8 9When the underlying set, S , of our dy-namical system, ( S, f ), has only a few ele-ments, it is convenient to describe our sys-tem as a directed graph . As an example, let S = { , , . . . , } be the first nine naturalnumbers. Let us arrange S as a square gridand consider the dynamical system describedby the graph at right.Representing the system as a directedgraph makes it convenient to identify the or-bit types. Indeed, we see that the element‘1’ is a fixed point; there are two periodic or-bits/cycles: (4 ,
7) and (5 , , ,
8) of period two2nd four respectively. Additionally, both ‘2’and ‘3’ are preperiodic points as they eventu-ally fall into the (5 , , ,
8) periodic orbit.
The mathematical notion of a field is as a place to do arithmetic. We recallthe definition of field below; a particular consequence is that the product of twononzero elements is again nonzero.
Definition 1.
A field is a set F equipped with two associative and commutativebinary operations + and × , such that • There exist elements and in F which are the identity elements for + and × respectively; • Every element a of F has an additive inverse: an element − a such that a + ( − a ) = 0 ; • Every nonzero element a of F has a multiplicative inverse: an element a − such that a × a − = 1 ; • The operation × distributes over + : a × ( b + c ) = a × b + a × c . Several fields are familiar to us, e.g., the real numbers R or the complexnumbers C . After some thought, one can see that the rational numbers (fractionsof integers with nonzero denominator) also has the structure of a field. Incontrast, note that the integers do not form a field: given a nonzero integerother than 1 or -1, say 5, there does not exist another integer with which theresulting product is equal to 1: 1 / { , , , , } equipped with the standard operation of addition, +, and multi-plication, × , except that we reduce modulo division by 5; let us label thesenew operations as e + and e × . That is, given two such integers, say 2 and 4, weconsider their standard product, but then only remember the remainder afterdividing by 5. More explicitly, 2 × e × e +4 = 1.) Equipped with the operations e +and e × , the set { , , , , } defines a field, we will denote this field F . Moregenerally, given any prime number p , there is a field F p with underlying set { , , , . . . , p − , p − } , as we can do arithmetic modulo any prime p !More generally, given any power of a prime, there exists a correspondingfinite field. That is, let n be a natural number and p a prime number, thereexists a unique field with p n elements, we denote this field F p n , e.g., F for n = 3and p = 2 or F for n = 2 and p = 3. We outline the existence and uniqueness ofthe field F p n at the end of the article, but let us describe explicitly our examplesof F and F . 3 xample 1. We present the field F as certain collection of polynomials. As aset, let F = { , , x, x + 1 , x , x + 1 , x + x, x + x + 1 } . The operations + and × are the standard sum and multiplication of polynomials,however, we must reduce all coefficients modulo 2, and use the replacement rulewhich sends x to x + 1 . For instance, we compute the following ( x + 1) × ( x + x + 1) = x + x + x + x + x + 1= x + 2 x + 2 x + 1= x + 1 + 2 x + 2 x + 1= 2 x + 3 x + 2= 0 + x + 0= x Example 2.
The field F has underlying set { , , , x, x + 1 , x + 2 , x, x +1 , x + 2 } . The operations + and × are again those of polynomials. We nowreduce all coefficients modulo 3 and use the replacement rule that x ≡ x + 1 . It is natural to wonder if there is a field of order n for any natural number n ...the answer is a resounding NO! Consider n = 6, it is the case that thestructure with respect to the operation + is that of addition modulo 6. Onethen sees that 2 × ≡
0, so there are nontrivial elements whose product is zero,which would contradict our earlier observation about fields.Given an (algebraic) equation, one could ask if it has solutions in a givenfield. For instance, the equation x − R , but foursolutions in C . Note that x − F , while thereare four solutions over F : 1, 2, x + 2, and 2 x + 1.Counting the number of solutions of equations over various fields is a ba-sic motivation for much of number theory and algebraic geometry. There areimportant functions which count the number of solutions of an equation overvarious fields, these are called Zeta functions . Zeta functions occupy a centralrole in many important theorems and conjectures: Hasse’s Theorem, the WeilConjectures, the Riemann Hypothesis, etc; we offer some explicit questions inSection 5.
We now come to the main focus of this note: dynamical systems determinedby polynomial functions. Our dynamical system will have as underlying set theelements of a finite field F p n . Given a polynomial, f ( t ), with integer coefficients,we can interpret f as a map from F p n to itself. More explicitly, given an element α of F p n we can evaluate f ( α ), apply the replacement rule defining F p n , andthen reduce all coefficients modulo p . It is more concrete to consider severalexamples, as we now do.Consider the polynomial f ( t ) = t + 1 over the field with nine elements, F .The directed graph below describes the dynamical system generated by iterating f ( t ). 4 1 2 x + 2 2 x + 1 2 xx + 1 x x + 2This dynamical system has one fixed point, a cycle of length two, and six prepe-riodic points.We now work with the same polynomial, f ( t ) = t + 1, but over the field F .0 x x + 1 x + x x + x + 1 x x + 1Note that there are no preperiodic points, no fixed points, and each element iscontained in one of two periodic orbits! Hence, it is clear that the dynamics ofiterating a polynomial depends significantly on the field. In the second example,the lack of preperiodic points is no coincidence, but rather a consequence of someelementary number theory as we will see in the next section.5 Guaranteeing no preperiodic points
In this section we prove an elementary, but useful result about polynomial dy-namics over finite fields. The key technical tool is the following result, typicallyattributed to Fermat and its extension that the Frobenius map is an automor-phism of a finite field.
Fermat’s Little Theorem.
Let p be a prime. Then for any integer a , a p ≡ a (mod p ) . Fermat’s result is incredibly useful, and it’s refinement due to Euler, is oneof the key ingredients of the RSA (Rivest-Shamir-Adleman) encryption scheme(also known as public-key cryptography). Though first described in 1977, theRSA algorithm, with some technical advances and padding , remains the mostwidely used encryption scheme for secure communication and digital commercein the world.There is an extension of Fermat’s Little Theorem which will prove useful.To begin, given a finite field F p n , there is a natural field homomorphism, the p th power map (also called the Frobenius endomorphism ): φ : F p n → F p n , a a p . That the map φ is actually a field homomorphism follows from the computations( xy ) p = x p y p and ( x + y ) p = x p + y p , with the latter a consequence of the Binomial Theorem.Before proving that the Frobenius map is actually an automorphism, weneed a standard result about field homomorphisms. Lemma 1.
Let ψ : G → H be a field homomorphism. Then ψ is injective.Proof. Let x, y ∈ G such that f ( x ) = f ( y ). Define an element d = x − y , so f ( d ) = f ( x − y ) = f ( x ) − f ( y ) = 0 . We will show d = 0. By way of contradiction, suppose not. So d = 0, then1 = f (1) = f ( dd − ) = f ( d ) f ( d − ) = 0 f ( d − ) = 0 , which is a contradiction in the field H . Proposition 1.
The Frobenius endomorphism φ : F p n → F p n is an automor-phism, in particular, φ is injective.Proof. Since F p n is finite, any injective map will automatically be surjective,hence its enough to prove that φ is injective. That φ is injective is the contentof the previous lemma. 6 heorem 1. Let p be a prime number, m a natural number, and c an element of F p . For all natural numbers n , the dynamical system generated by f ( t ) = t p m + c has no preperiodic points over F p n .Proof. Note that our finite system will have no preperiodic points when themap f is injective, hence we prove injectivity of f . Suppose a, b ∈ F p n suchthat a p m + c = b p m + c . By cancellation with respect to +, we then know that a p m = b p m .We now induct on m . In the case m = 1, we deduce that a = b by theproceeding proposition. Now assume that for 1 ≤ k ≤ N , if a p N = b p N , then a = b in F p n . We consider the case where k = N + 1. Suppose, that a p N +1 = b p N +1 . Note that, a p N +1 = a p a p N and similarly, b p N +1 = b p b N +1 . Applying theinductive hypothesis, to a p N and b p N , we conclude a p = b p which is the basecase of our induction. Therefore, for all natural numbers m , we have proventhat if a p m = b p m , then a = b and we have established the injectivity of f .Note that the lack of pre-periodic points does not imply that our system isof the form of the proposition. Indeed, over F the polynomial f ( z ) = z onlyhas fixed points. There are many proofs of Fermat’s Little Theorem and nearly every text onnumber theory contains at least one such proof. Euler gave a proof of the theo-rem based on the Binomial Theorem; there are also proofs that use basic grouptheory and a proof due to Ivory and Dirichlet that uses modular arithmetic.Motivated by the connection to dynamical systems, we recall a proof of thetheorem due to Iga [5] (see also [2]).Let n be an integer, Iga’s idea is to study the dynamics on the interval [0 , T n ( x ) = (cid:26) nx − ⌊ nx ⌋ , x = 11 , x = 1Note that y − ⌊ y ⌋ is just the factional part of the real number y .The reader is encouraged to draw a graph of the function T n ( x ), which willmake the following clear. Proposition 2.
Let ℓ, m, n be integers with n > . Then,(a) T n ( x ) has n fixed points;(b) ( T m ◦ T ℓ )( x ) = T mℓ ( x ) . Let a be an integer and n a prime number. We will now use the dynamicsdetermined by T a to show that p divides the difference a p − a , which is equivalentto Fermat’s Little Theorem. 7ndeed, let P denote the set of points which have period p with respect toiterating T a . Equivalently, P is the set of fixed points of T a p , so |P| = a p . Ofthese a p fixed points, a are already fixed by T a . As p is prime, we know that a p − a points then have minimal period p , and hence the number of orbits oflength p is exactly ( a p − a ) /p . But then, as a finite count of orbits, ( a p − a ) /p must be an integer; that is p divides a p − a . Elliptic curves are geometric objects that can be defined over arbitrary fields.An elliptic curve, E , defined over the complex numbers C has a particularlynice description: E is a quotient of C by a lattice L = Z ⊕ Z τ for some complexnumber τ in the upper half plane. This description makes it clear that an ellipticcurve over C is, as a space, simply a torus. Elliptic curves are also examplesof algebraic varieties , in that they are the space of solutions to an algebraicequation. Indeed, our curve E can be described as the space of solutions to acubic equation in two variables: y + a xy + a y = x + a x + a x + a . The equation above is called the (generalized) Weierstrass equation. Given anelliptic curve, E , we can ask how many points this curve has over the finitefield F q . Equivalently, this is the number of solutions to the correspondingWeierstrass equation in F q ; denote this number N ( q ). It is a classical theoremof Hasse that | N ( q ) − q − | ≤ √ q. The collection of periodic points in a dynamical system determined by iter-ating a polynomial (even rational) map also has the structure of an algebraicvariety, see [8]. This description was extended in [4] to the case of pre-periodicpoints. Hence, one could ask the following.(Q1) Can one bound the number of pre-periodic points in analogy with Hasse’sTheorem?For a fixed prime p , one can assemble the numbers N ( p m ) into a formal powerseries (with further coefficients), called the (local) zeta function . (Again, see [8]for more.) For algebraic curves, this power series is analogous to the RiemannZeta Function which is the subject of the famous Riemann Hypothesis. Thenature of the zeta function of an algebraic curve led to some of the deepest andencompassing mathematics of the 20th century: The Weil Conjectures.Pre-periodic zeta functions have only been lightly studied, e.g., [9] and thereare many aspects to be explored.(Q2) What can one determine about the pre-periodic zeta function?8y “pasting together” the local zeta functions over each prime, one formsthe Dedekind Zeta Function. In good situations, one can read off a group ofsymmetries from the Dedekind Zeta Function: the Galois group. The case ofperiodic points has been studied in [7].(Q3) Can one compute an analogue of the Galois group for pre-periodic vari-eties? Appendix: Existence and Uniqueness of FiniteFields
We now prove that given a prime number p and a natural number n , up toisomorphism, there is a unique field with p n elements. Hence, our notation F p n is unambiguous (up to isomorphism). The proof of the theorem uses somegraduate level abstract algebra that may not be familiar to many readers, astandard reference is [3]. Theorem 2.
Let p be a prime and n a natural number. Then, • (Existence) There exists a field with p n elements. • (Uniqueness) All fields with p n elements are isomorphic.Proof of Existence. Consider the ring of polynomials with integer coefficients Z [ x ]. Let p ( x ) be an irreducible polynomial of degree n . As Z is a principalideal domain, the ideal generated by p ( x ) in Z [ x ] is maximal, so Z [ x ] / ( p ( x )) isa field. It is straightforward to verify that Z [ x ] / ( p ( x )) has p n elements. Sketch of Proof of Uniqueness.
The main idea is to consider the splitting field K of the polynomial x p n − x over F p . This splitting field has p n elements byminimality. Now the group of units F × p n is a cyclic group of order p n − x p n − −
1. Hence, F p n is a splitting field of x p n − x over F p . By uniqueness of splitting fields, we have that F p n ∼ = K . Theorem 3.
Let n ∈ N such that there exist distinct primes p and q bothdividing n . There exists no field of order n . The theorem follows from a simpler proposition.
Proposition 3.
Let p and q be distinct primes, then there exists no field oforder pq .Proof. Suppose F is a field of order pq and let 1 denote the unit with respectto multiplication × . One can show that the underlying abelian group ( F , + , Z /pq (an application of the Sylow Theorems).Let ρ = p × δ = q ×
1. Then, ρ × δ = pq × F . That is, ρ (and δ ) is a nontrivial zero divisor, contradicting F being afield. 9 eferences [1] Robert Benedetto, Laura DeMarco, Patrick Ingram, Rafe Jones, MichelleManes, Joseph H. Silverman, Thomas J. Tucker. Current Trends and OpenProblems in Arithmetic Dynamics . https://arxiv.org/abs/1806.04980 [2] Dragovi´c, Vladimir. Polynomial dynamics and a proof of the Fermat littletheorem.
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About the authors:
Ryan Grady
Ryan Grady is an Assistant Professor of Mathematics and Director of the Di-rected Reading Program in Mathematical Sciences at Montana State.Department of Mathematical Sciences, Montana State University, Bozeman59717. [email protected]