Urn Nbn Se Vxu Diva-403-1 Fulltext

Monomial Dynamical Systems in the Fields of p-adic Numbers and Their Finite Extensions

Acta Wexionensia No 62/2005 Mathematics

Monomial Dynamical Systems in the Fields of p-adic Numbers and Their Finite Extensions

Marcus Nilsson

Växjö University Press

Monomial Dynamical Systems in the Fields of p-adic Numbers and Their Finite Extensions. Thesis for the degree of Doctor of Philosophy, Växjö University, Sweden 2005

Series editors: Tommy Book and Kerstin Brodén ISSN: 1404-4307 ISBN: 91-7636-458-5 Printed by: Intellecta Docusys, Göteborg 2005

Abstract In this thesis we investigate monomial dynamical systems over the fields of p-adic numbers, Qp , or over finite field extensions of them. A finite extension of Qp is denoted by Kp . The monomial dynamical systems are described by iterations of h(x) = xn , where n
2 is an integer. Since the fields Kp are not algebraically closed, the number of r-periodic points, Pr (h(x), Kp ), of h(x) in Kp will vary very much with p. We find explicit formulas for obius Pr (h(x), Kp ), by using methods from number theory, for example M¨ inversion. One of the main parts of this thesis is the computation of the limit 1
Pr (h, Kp ), lim t→∞ π(t) p
t

where π(t) denotes the number of primes p  t and where the sum is extended over all prime numbers p  t. We interpret this limit as the asymptotic mean value of the number of r-periodic points of h(x) in Kp , when p → ∞. For this to make sense, we must assume that the degree of the extension Kp /Qp is the same for all p. We also study the dynamics of balls in Qp under the monomial h(x). We will call a cycle of balls a fuzzy cycle. Methods for calculating the number of fuzzy cycles are presented. In this thesis we also consider perturbed monomial systems over the field of p-adic numbers. This systems are generated by polynomials hq (x) = xn + q(x), where the perturbation q(x) is a polynomial whose coefficients have a small p-adic absolute value. We find sufficient conditions on the perturbation to guarantee a one to one correspondence of fixed points and cycles between the monomial and the perturbed system. Keywords: p-adic numbers, discrete dynamical systems, number of cycles, perturbation, roots of unity, M¨ obius inversion, distribution of prime numbers.

v

Acknowledgements First of all, I would like to thank my supervisor Andrei Khrennikov for introducing me to the subject of p-adic dynamical systems, and for his inspiring ideas and advices. I also want to thank my co-supervisor Anders Melin, University of Lund. Scientific contacts with him were very important during the writing of my master thesis and at the initial stages of my graduate studies. I thank Alain Escassut, Bertin Diarra at Universit´e Blaise Pascal in Clermont-Ferrand, and Nicolas Mainetti at Universit´e d’Auvergne for their hospitality and the fruitful discussions. I would also like to thank Franco Vivaldi at Queen Mary University of London and Robert Benedetto at Amherst College for comments, ideas and suggestions of improvments. I thank my colleagues, Karl-Olof Lindahl, Robert Nyqvist and Per-Anders Svensson, in the research group in p-adic dynamics, for many interesting seminars and discussions. I also thank Robert Nyqvist for answering my questions about unix, emacs and latex. I would also like thank my other colleagues at the School of Mathematics and Systems Engineering at V¨ axj¨ o University, for creating such a nice working environtment. During the writing of this thesis both my parents died. I owe them a large dept of gratitude. Without their support and encouragement I would never have started my graduate studies. Finally, I thank my wonderful family, my beloved Malin and our daughter Clara, for love, patience and encouragement.

vi

Contents Abstract

v

Acknowlegements

vi

General Introduction 1 Introduction 2 Fields of p-adic Numbers 2.1 Non-Archimedean fields . . . . . . . . . 2.2 The field of p-adic numbers . . . . . . . 2.3 Extensions of the field of p-adic numbers 2.4 Hensel’s lemma . . . . . . . . . . . . . . 2.5 Roots of unity . . . . . . . . . . . . . . . 3 Discrete Dynamical Systems 3.1 Periodic points and their character . . . 4 Summary of the Thesis References

1 3 4 4 5 7 9 9 11 11 13 15

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I Cycles of Monomial and Perturbed Monomial p-adic Dynamical Systems 19 1 Introduction 21 2 Properties of Monomial Systems 21 3 Number of Cycles 23 4 Distribution of Cycles 27 5 Existence of Fixed Points of a Perturbed System 32 6 Cycles of Perturbed Systems 37 References 43 II Distribution of Cycles of Monomial p-adic Dynamical Systems 45 1 Introduction 47 2 Notation and Earlier Results 47 3 Cycles of Monomial Systems 48 4 Distribution of Cycles 50 5 Expectation and Variance of ξ 54 6 Acknowlegement 56 References 56 III Asymptotic Behavior of Periodic Points of Monomials in the Fields of p-adic Numbers 59 1 Introduction 61 2 Roots of Unity 61 2.1 Roots of unity in Fp . . . . . . . . . . . . . . . . . . . . . . . 62 2.2 Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . 63 3 Periodic Points and Cyclotomic Polynomials 64 4 Group Action on Finite Sets 67 5 Periodic Points in Finite Fields 68 vii

6 Up to Qp 69 6.1 The p-adic fields . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.2 Roots of unity in Qp . . . . . . . . . . . . . . . . . . . . . . . 71 6.3 Periodic points in p-adic fields . . . . . . . . . . . . . . . . . . 72 References 73 IV Fuzzy Cycles of p-adic Monomial Dynamical Systems 75 1 Introduction 77 2 Global Dynamics 79 3 Local Dynamics 82 3.1 Dynamics around neutral points . . . . . . . . . . . . . . . . 85 3.2 Dynamics around attractors . . . . . . . . . . . . . . . . . . . 92 4 Distribution of Fuzzy Cycles 93 References 94 V Monomial Dynamics in Finite Extensions of the Fields of p-adic Numbers. 95 1 Introduction 97 2 Finite Field Extensions of the Field of p-adic Numbers 97 3 Roots of Unity 98 4 Monomial Dynamics 100 5 Counting Periodic Points 101 6 Asymptotic Behavior 103 7 Periodicity 106 References 109

viii

General Introduction

1

1

Introduction

Introduction

Almost everything in nature and society that evolves in time can be described as a dynamical system, the solar system, the weather, the stock exchange, the flow of minds in our brains, et cetera. We essentially have two kinds of dynamical systems, continuous and discrete. A continuous dynamical system describes phenomena that evolve continuously in time, while a discrete dynamical system describes phenomena that evolve only at certain moments. In the latter case the dynamical system is described by iterations of a function. In this thesis we will consider discrete dynamical systems. The purpose of studying dynamical systems is to predict the future of a given phenomenon. Even if we could not do this exactly we are often able to get some information about the long-time behavior of the system. One way to classify this behavior is to find fixed points and cycles of the system, and determine if these are attractive, repulsive or neutral. When constructing a model of a phenomenon in nature or in society we, almost without exception, use some set of numbers to represent a measurable quantity of the phenomenon. The by far most used set of numbers is the field of rational numbers, Q, or some field that contains them (for example the real numbers or the complex numbers). We also need a way of measuring distances between different numbers, that is, we need a metric or a topology on the set of numbers we use. On the rational numbers we very often use the absolute value of the difference of two numbers to measure the distance between them. But there are many other possibilities, for example the so called p-adic absolute value. We know that Q is not complete as a metric space with the metric induced by the ordinary absolute value. We have the same situation for Q endowed with the metric ρ induced by the p-adic absolute value. The completion is the field of p-adic numbers, denoted by Qp . The metric ρ satisfies the strong triangle inequality ρ(x, z)  max(ρ(x, y), ρ(y, z)),

(1.1)

and Qp is therefore an example of a so called ultrametric space. The fact is that the ordinary absolute value and the p-adic absolute values (for each prime number p) are essentially the only absolute values on Q. This result is described in Ostrovski’s Theorem, see for example [14, 17, 32]. This theorem gives the p-adic numbers a special position, when it comes to modeling. Which way of measuring distances on Q we choose must depend on the phenomenon we are making a model of. Of course, the most common way of measuring distances is by the ordinary absolute value, but there are phenomena that need the p-adic distance, see for example [35, 16, 17, 36]. In this thesis we will study discrete dynamical systems over the Qp and over finite field extensions of Qp . During the last couple of dekades there has been an increasing interest for p-adic dynamical systems, induced by 3

General Introduction

p-adic mathematical physics, see [35, 36, 4, 13, 16] 1 . The p-adic dynamical systems have for example been studied in [17, 21, 23, 33, 34] and in [5, 8, 9, 15, 22, 29, 30]. There were studies not only on dynamical systems over Qp but also on extensions of Qp and more general non-Archimedean fields. There are also several articles, [6, 7, 12, 20, 19], that propose p-adic models for cognitive processes. Recently, in [1], multidimensional non-Archimedean dynamical systems have been investigated. The aim of this thesis is to investigate and describe a dynamical systems given by iterations of the monomial h(x) = xn , n
2.

(1.2)

Even though this is function of a very simple type, we will see that the dynamics will have a rich structure. A detailed analysis of monomial systems over non-Archimedean fields (in particular Qp ) was first provided in [17]. Monomial systems over the field Qp and its finite exensions have then been studied in for example [24], [25], [26], [27] and in [28]. These are the articles that this thesis is based upon. There are essentially two things that make the monomial dynamics more interesting in Qp and its finite field extensions, than in the field C of complex numbers. First, Qp and its finite extensions are not algebraically closed. For exampel, the number of periodic points of a fixed period will vary with p. Second, Qp and its finite extensions are totally disconnected, and they possess a tree structure. Hopefully the structures described in this thesis will imply new areas of applications of p-adic dynamical systems in the future.

Fields of p-adic Numbers

2

In this chapter we will give a short introduction to the fields of p-adic numbers and their extensions.

2.1

Non-Archimedean fields

Definition 2.1. Let K be a field. An absolute value on K is a function |.| : K → R such that • |x|
0 for all x ∈ K, • |x| = 0 if and only if x = 0, • |xy| = |x||y|, for all x, y ∈ K, • |x + y|  |x| + |y|, for all x, y ∈ K. 1 p-adic mathematical physics also stimulated studies in non-Archimedean functional analysis, see [10, 11, 2, 3].

4

2

Fields of p-adic Numbers

If |.| in addition satisfies the strong triangle inequality, |x + y|  max(|x|, |y|),

(2.1)

for all x, y ∈ K then we say that |.| is non-Archimedean. If |x| = 1 for all non-zero x ∈ K we call |.| the trivial absolute value. It is easy to see that the trivial absolute value is non-Archimedean. Proposition 2.2. Let K be a field and let |.| be a non-Archimedean absolute value on K. Let x, y ∈ K such that |x| = |y|. Then |x + y| = max(|x|, |y|).

(2.2)

Every non-Archimedean field can be regarded as an ultrametric space with the metric ρ(x, y) = |x − y| induced by the absolute value. Let a ∈ K and let r ∈ R+ . The open ball of radius r with center a is the set Br− (a) = {x ∈ K : |x − a| < r}. The closed ball of radius r with center a is the set Br (a) = {x ∈ K : |x − a|  r}. The set Sr (a) = {x ∈ K : |x − a| = r} is called the sphere of radius r with center a. It is sometimes important to underline in which field a ball or a sphere is included. We then use the symbols Br− (a, K), Br (a, K) and Sr (a, K). The strong triangle inequality and Proposition 2.2 have some remarkable consequences for the balls in K. • Every element of a ball can be regarded as a center of it. • Each open ball is both open and closed as sets. • Each closed ball of positive radius is both open and closed. • Let B1 and B2 be balls in X. Then either B1 and B2 are orderd by inclusion (B1 ⊆ B2 or B2 ⊆ B1 ) or B1 and B2 are disjoint. • An ultrametric space is totally disconnected.

2.2

The field of p-adic numbers

Let p be a fixed prime number. By the fundamental theorem of arithmetics, each non-zero integer n can be written uniquely as n = pvp (n) n , 5

General Introduction

where n is a non-zero integer, p
n , and vp (n) is a unique non-negative integer. The function vp : Z \ {0} → N is called the p-adic valuation. If a, b ∈ Z+ then we define the p-adic valuation of x = a/b as vp (x) = vp (a) − vp (b).

(2.3)

One can easily show that the valuation is well defined. The valuation of x does not depend on the fractional representration of x. By using the padic valuation we will define a new absolute value on the field of rational numbers. Definition 2.3. The p-adic absolute value of x ∈ Q \ {0} is given by |x|p = p−vp (x)

(2.4)

and |0|p = 0. When it is clear from the context which absolute value we use we denote the p-adic absolute value by |.|. It is easy to prove that the p-adic absolute value is non-Archimedean, and that the metric ρ(x, y) = |x − y|p induced by it, is an ultrametric. Two absolute values on a field K are said to be equivalent if they generate the same topology on K. Essentially there are only two types of non-trivial absolute values on Q. This is the essence of the following theorem. Theorem 2.4 (Ostrovski). Every non-trivial absolute value on Q is either equivalent to the real absolute value or to one of the p-adic absolute values. For a proof of Ostrovski’s theorem see, for example, [32] or [14]. Let Q be endowed with the ultrametric induced by the p-adic absolute value. However, this space is not complete. The completion of Q will be a field, the field of p-adic numbers, Qp . The p-adic absolute value can be extended to Qp and Q is dense in Qp . It is worth noting that {|x|p : x ∈ Qp } = {|x|p : x ∈ Q} = {pm : m ∈ Z} ∪ {0}. The set B1 (0) = {x ∈ Qp ; |x|p  1} is sometimes called the p-adic integers. It is denoted by Zp . In fact, Zp is a subring of Qp and B1− (0) = {x ∈ Zp ; |x|p < 1} is a maximal ideal of Zp . The quotient ring Zp /B1− (0) is then a field, called the residue class field of Qp . The residue class field of Qp is isomorphic to the finite field Fp of p elements. Theorem 2.5. Every x ∈ Qp can be expanded in the base p in the following way
yj pj , (2.5) x= jjmin

where jmin = vp (x) ∈ Z and 0  yj  p − 1 for j
jmin . 6

2

2.3

Fields of p-adic Numbers

Extensions of the field of p-adic numbers

Everywhere below we denote by Kp a finite extension of the p-adic numbers. Let m = [Kp : Qp ] denote the dimension of Kp as a vector space over Qp . The p-adic absolute value |.|p can be extended to Kp , in the unique way. See [14], [32] or [31] for details. But, how can we evaluate the p-adic valuation on elements in Kp ? We need a function NKp /Qp : Kp → Qp , which satisfies the equality NKp /Qp (xy) = NKp /Qp (x) NKp /Qp (y). This function is the so called norm from Kp to Qp . There exists several ways to define NKp /Qp , all equivalent. Below, three of them are listed. • Let α ∈ Kp and consider Kp as a finite dimensional Qp -vector space. The map from Kp to Kp defined by multiplication by α is a Qp -linear map. Since it is linear it corresponds to a matrix. We define NKp /Qp to be the determinant of this matrix. • Let α ∈ Kp and consider the subfield Qp (α). Let r = [Kp : Qp (α)] and let T (α, Qp ) be the minimal polynomial of α over Qp and let n = deg(T (α, Qp )). Then the norm is defined as NKp /Qp (α) = (−1)nr ar0 , where T (α, Qp ) = an xn + an−1 xn−1 + · · · + a1 x + a0 . • Suppose that Kp is a normal extension of Qp . Let G(Kp /Qp ) be the Galois group of this extension. Then, for α ∈ Kp , the norm is defined as  σ(α), for all σ ∈ G(Kp /Qp ). NKp /Qp (α) = Observe that |G(Kp /Qp )| = [Kp : Qp ], because Kp is a normal extension of Qp and Qp is of characteristic zero. Since NKp /Qp (α) ∈ Qp for each α ∈ Kp it has a p-adic absolute value. We can now use this to extend the p-adic absolute value to Kp . Theorem 2.6. Let Kp be a finite extension of Qp and m = [Kp : Qp ]. Then the function |.| : Kp → R+ defined by |x| =

 | NKp /Qp (x)|p

m

is a non-Archimedean valuation on Kp that extends |.|p . 7

General Introduction

Since |.| is unique, |.|p can also be used to denote the extended p-adic valuation. From algebra we know that for each finite extension Kp of Qp there exists a finite normal extension of Qp which contains Kp . The smallest such normal extension of Qp is called the normal closure of Qp over Kp . If Kp is not a normal extension of Qp and we want to define a norm by using Qp -automorphisms, then we consider the normal closure of Qp over Kp and use the third definition of the norm. Let Kp be a finite field extension of Qp and m = [Kp : Qp ]. For x ∈ Kp set y = NKp /Qp (x). Then we have by Theorem 2.6 that |x|p =

  m |y|p = p−vp (y) = p−vp (y)/m = p−vp (x) ,

m

1 where vp (x) = vp (y)/m, that is, vp (x) ∈ m Z, because vp (y) ∈ Z. If a, b ∈ Kp then vp (ab) = vp (a) + vp (b). This gives that vp is a homomorphism from the multiplicative group K× p to the additive group Q. Then 1 Z. Let the image Im(vp ) is an additive subgroup of Q, and Im(vp ) ⊆ m d/e be in Im(vp ), where d and e are relatively prime, chosen so that the denominator e is the largest possible. This choice can be done because e has to be a divisor of m, and the set of possible divisors is bounded. Since d and e is relatively prime, there must be a multiple of d which is congruent to 1 modulo e, that is, we can find r and s such that rd = 1 + se. But then

r

1 + se 1 d = = +s e e e

1 Z, it follows that 1/e ∈ Im(vp ). Since e is in Im(vp ). Since s ∈ Z ⊂ m was chosen to be the largest possible denominator in Im(vp ), it follows that Im(vp ) = 1e Z. This unique positive integer e is called the ramification index of Kp over Qp . The extension Kp over Qp is called unramified if e = 1, ramified if e > 1 and totally ramified if e = m.

Definition 2.7. We say that an element π ∈ K is a uniformizer if vp (π) = 1/e. The unit ball B1 (0, Kp ) = {x ∈ Kp ; |x|  1} is a subring of Kp and B1− (0, Kp ) is a unique maximal ideal of B1 (0, Kp ). The quotient ring is then a field, the residue class field of Kp . We state a few facts about the extention Kp : • Kp is locally compact and complete, but it is not algebraically closed • Each x ∈ Kp can be written as x = uπ vπ (x) , where u ∈ S1 (0) and vπ (x) = vp (x)e. • The residue class field of Kp is isomorphic to Fpf , the field of pf elements, where f = m/e. This number is called the residue class field degree. The residue class field of Qp is Fp so f is the degree of the residue class field as an extension of the residue class field of Qp . 8

2

Fields of p-adic Numbers

• Let C = {c0 , c1 , . . . , cpf −1 } be a fixed complete set of representatives of the cosets of B1− (0) in B1 (0). Then every x ∈ Kp has a unique π-adic expansion of the form
x= = ai π i , ii0

where i0 ∈ Z and ai ∈ C for every i
i0 . The union of all finite extentions of Qp is a field and an algebraic closure of Qp . We denote this field by Qp . It is possible to extended the p-adic absolute value to Qp . The possible positive values are pr , where r ∈ Q. The algebraic closure Qp of Qp is an infinite extension and it is not complete with respect to the metric induced by the p-adic absolute value. The completion of Qp is however also algebraically closed and we call it the field of complex p-adic numbers. We denote this field by Cp .

2.4

Hensel’s lemma

Let Kp be a finite extension of Qp and let π be a uniformizer. Let α, β ∈ Kp . We say that α ≡ β (mod π γ ) if |α−β|p  |π|γp . The following theorem and its corollary are important tools for finding solutions of polynomial equations. Theorem 2.8. Let F (x) be a polynomial with coefficients in B1 (0, Kp ). Assume that there exists α0 ∈ B1 (0, Kp ) and γ ∈ N such that F (α0 ) ≡ 0 (mod π 2γ+1 ) F  (α0 ) ≡ 0 (mod π γ ) F  (α0 ) ≡ 0 (mod π γ+1 ). Then there exists α ∈ B1 (0, Kp ) such that F (α) = 0 and α ≡ α0 (mod π γ+1 ). Corollary 2.9 (Hensel’s lemma). Let F be a polynomial with coefficents in B1 (0, Kp ) and suppose that there exits α0 ∈ B1 (0, Kp ) such that F (α0 ) ≡ 0 (mod π) and F  (α0 ) ≡ 0 (mod π). Then there exists α ∈ B1 (0, Kp ) such that F (α) = 0 and α ≡ α0 (mod π).

2.5

Roots of unity

The roots of unity in Kp will be essential for our investigations of the monomial dynamical systems. Let Kp be a finite extension of Qp of degree m = e · f , where e is the ramification index and f is the residue class degree. The residue class field is isomorphic to Fpf . Definition 2.10. We say that x ∈ Kp is an n-th root of unity f xn = 1. If xn = 1 and xm = 1 for every m < n then we say that x is a root of unity of order n or a primitive n-th root of unity. 9

General Introduction

The multiplicative group of Fpf is cyclic and has pf − 1 elements. Since a cyclic group has a cyclic subgroup of order d for each divisor d of pf − 1, for every d | pf − 1 there exists x ∈ F× that generates the subgroup of d pf elements and we also have xd = 1. The element x generates a group of d roots of the polynomial xd − 1 in Fpf . Let us denote the d roots x1 , . . . , xd . Take now d elements y1 , . . . yd of S1 (0, Kp ) such that yj belongs to the coset that corresponds to xj . Then there are d approximate roots of F (x) = xd − 1 = 0 in B1 (0, Kp ) because F (yj ) ≡ 0 (mod π) and F  (yj ) ≡ 0 (mod π). Of course, the d different yj are located in d different cosets of B1− (0, Kp ). Hence they are noncongruent modulo π. By Hensel’s lemma, for each d | pf − 1, the equation xd − 1 = 0 has d solutions in Kp . We have proved the following proposition. Proposition 2.11. The field Kp contains the (pf − 1)-roots of unity. We also have the following results, see for exemple [14] or [31] for proofs. Proposition 2.12. Let n be an integer that is relatively prime to pf − 1. Let xn = 1. Then x ≡ 1 (mod π) or in other words x ∈ B1− (1). Proposition 2.13. If x ∈ B1− (1) such that xn = 1 then n is divisible by a power of p and x is a root of unity for that power of p. We know even more about the p-power roots of unity. Theorem 2.14. Let ζ be a pt th root of unity in Kp . Then |ζ − 1|p = 1/ϕ(pt )

|p|p

, where ϕ(pt ) = pt−1 (p − 1) (Euler’s ϕ-function).

See [31] for a proof. We have the following immediate consequences of this theorem. Corollary 2.15. Let e be the ramification index of Kp as an extension of Qp . Let ζ ∈ Cp be a root of unity of order pt , where t
1. A neccesary condition for ζ ∈ Kp is that ϕ(pt ) | e. Corollary 2.16. Assume that Kp has ramification index e as an extension of Qp . We then have at most e/(1 − 1/p) p-power roots of unity. Corollary 2.17. Let m be the degree of Kp as an extension of Qp . Then there is only a finite number of primes p, such that Kp possesses a p-power root of unity. Proof. Let e be the ramification index of the extension Kp /Qp . If p − 1 > m then p − 1 > e and ϕ(ps )
e for any s
1. Let tˆ be the largest integer for which there exists a root of unity of order ptˆ in Kp . Recall that if ζ is a root of unity of order pt , then ζ generates a cyclic group of order pt . The elements of this group are of course p-power roots of unity of order ps , where 0  t  t. 10

3

Discrete Dynamical Systems

Theorem 2.18. Let 1 < t  tˆ On the sphere Sp−1/ϕ(pt ) (1) there are ϕ(pt ) different roots of unity, all of order pt . Moreover, Kp contains ptˆ, p-power roots of unity. Proof. Let ζ be a root of unity of order ptˆ. Since the group generated by ζ is cyclic, where are cyclic groups of order ps for every 1 < s  tˆ. Every such group has ϕ(ps ) generators, which all are roots of unity of order ps . Since tˆ t tˆ tˆ t=0 ϕ(p ) = p , Kp contains p p-power roots of unity. The rest follows from Theorem 2.14. Theorem 2.19. Let Kp be a finite extension of Qp with residue class degree f . Let tˆ be the p-power root of unity of highest order. We then have (pf −1)ptˆ roots of unity in Kp .

3

Discrete Dynamical Systems

This chapter is devoted to discrete p-adic dynamical systems, namely iteration (3.1) xn+1 = h(xn ) of functions h : K → K on a complete non-Archimedean field K. Mostly, we will let K be Qp or a finite extension, Kp , of Qp . Below, we will sometimes write “the dynamical system h(x)” when referring to the dynamical system that is described by iterations of h.

3.1

Periodic points and their character

For a given point x0 the set of points {hm (x0 ); m ∈ N} is called the trajectory or orbit through x0 . Some orbits of a dynamical system are of particular interest: Definition 3.1. A point x0 ∈ X is said to be a periodic point if there exists r ∈ N such that hr (x0 ) = x0 . The least r with this property is called the period of x0 . If x0 has period r, it is called an r-periodic point. A 1-periodic point is called a fixed point. The orbit of an r-periodic point x0 is {x0 , x1 , . . . , xr−1 }, where xj = hj (x0 ), 0  j  r − 1. This orbit is called an r-cycle. An r-cycle consists of r different r-periodic points. See Figure 3.1. Each element of the cycle has the cycle as its orbit. As a simple consequence we have that the number of r-periodic point of a discrete dynamical system is always divisible by r. The periodic points have different charcters. 11

General Introduction

f

x1 f

x2

x0

f x3

f f

x4

Figure 3.1: A 5-cycle contains five different 5-periodic points. Definition 3.2. Let x0 be an r-periodic point and let g(x) = hr (x). If there exists a ball Bρ− (x0 ) such that for every x ∈ Bρ− (x0 ) we have lim g s (x) = x0

s→∞

then we say that x0 is an attractor. The set A(x0 ) = {x ∈ X; lim g s (x) = x0 } s→∞

is called the basin of attraction of x0 . Definition 3.3. Let x0 be an r-periodic point and let g(x) = hr (x). If there exists a ball Bρ− (x0 ) such that |x − x0 | < |g(x) − x0 | for every x ∈ Bρ− (x0 ), x = x0 then x0 is said to be a repeller. Definition 3.4. See [17]. Let x0 be an r-periodic point. If there exists an open ball Bρ− (x0 ) such that for every ρ < ρ the spheres Sρ
(x0 ) are invariant under the map g = hr then Bρ− (x0 ) is said to be a Siegel disk and x0 is said to be a center of a Siegel disk. The union of all Siegel disks with center x0 is the Siegel disk of maximal radius of x0 . It is denoted by SI(x0 ). Definition 3.5. An r-periodic point x0 is said to be attractive if |g  (x0 )| < 1, indifferent if |g  (x0 )| = 1 and repelling if |g  (x0 )| > 1. Theorem 3.6. Let a be a fixed point of a dynamical system given by a polynomial function h(x) over a non-Archimedean valued field. (i) If a is an attracting point of h then it is an attractor of the dynamical system. 12

4

Summary of the Thesis

(ii) If a is an indifferent point of h then it is the center of a Siegel disk. (iii) If a is a repelling point of h then a is a repeller of the dynamical system. The following Theorem follows directly from the chain rule. Theorem 3.7. If one r-periodic point of an r-cycle is an attractor (repeller, center of a Siegel disc) then all the r-periodic points of that cycle are attractors (repellers, centers of Siegel discs). In view of this theorem, it makes sense to speak about the basin of attraction of a cycle. Definition 3.8. Let γ be an r-cycle, say  γ = {x0 , x1 , . . . , xr−1 }. The basin of attraction of γ is defined as A(γ) = x∈γ A(x), where A(x) is the basin of attraction of x.

4

Summary of the Thesis

We investigate monomial dynamical systems over Qp or over finite field extensions over Qp , denoted by Kp . The monomial dynamical systems are described by iterations of h(x) = xn , n ∈ N, n
2.

(4.1)

Even though this is function of a very simple type, we will see that the dynamics will have a rich structure. A detailed analysis of monomial systems over non-Archimedean fields, in particular over Qp , its finite extensions and the field of complex numbers p-adic numbers, Cp , was first provided in [17]. The field Qp and its finite extensions are not algebraically closed. For example, the number of r-periodic points will vary with p. This is one of the reasons why monomial dynamics are more interesting over Kp than over the real or complex numbers. Let Pr (h, K) denote the number of r-periodic points of h in the field K. In Paper I we find a formula for Pr (h, Qp ), by using methods from number theory, for example M¨ obius inversion. In Paper V this formula is generalized to finite extensions. We find that
ˆ µ(r/d) gcd(nd − 1, (pf − 1)pt ), (4.2) Pr (h, Kp ) = d|r

where µ is M¨obius function and tˆ is the largest integer such that there exists a root of unity of order ptˆ. If p > 2 and Kp = Qp then tˆ = 0 and f = 1. A large part of this thesis circulates around the limit 1
Pr (h, Kp ), t→∞ π(t) lim

(4.3)

p
t

13

General Introduction

where π(t) denotes the number of primes p  t and the sum is extended over prime numbers less than or equal to the real number t. We interpret this limit as the asymptotic mean value of the number of r-periodic points in Kp , when p → ∞. The limit (4.3) is calculated for Qp in Paper I, Paper II and Paper III by use of different methods. We find that
1
Pr (h, Qp ) = µ(r/d)τ (nd − 1), t→∞ π(t) lim

p
t

(4.4)

d|r

where τ (m) for m ∈ Z+ denotes the number of positive divisors of m. In Paper I, we perform an elementary proof by use of some well known technics from number theory. In Paper II we instead use probabilistic methods for proving (4.4). We run into some difficulties when defining a probability measure on the set of prime numbers, but solve this by considering a more general probability concept, see [18]. There is no uniform Kolmogorov probability measure on the set of all prime numbers. Therefore we have to consider a finite additive “generalized probability”. We consider the number of cycles as a generalized random variable and calculate the expectation and the variance with respect to a finite-additive probability measure. In Paper III we use Galois theory and methods from algebraic number theory, especially the ˇ Theorem of Cebotarev to compute the limit (4.4). In Paper V we instead consider the limit (4.3) for finite extensions Kp och Qp , with fixed residue class field degree f . By generalizing the technic from Paper I we find that

1
Pr (xn , Kp ) = µ(r/d) νf (l), t→∞ π(t) d lim

p
t

d|r

(4.5)

l|n −1

where νf (l) denotes the number of solutions of xf ≡ 1 (mod l). It turns out that the limit (4.5) is a periodic in f . In paper IV we study the dynamics of balls in Qp under the monomial x → xn . Following Khrennikov in [17] a cycle of balls is called a fuzzy cycle1 . There is a one-to-one correspondence between the fuzzy cycles of balls of radius 1/p and the cycles in Qp . However, the structure of fuzzy cycle of balls of radius r  1/p2 is non-trivial. Some numerical experiments to clearify the structure were performed in Khrennikov [17]. In Paper IV the structure of fuzzy cycles is investigated by analytic methods. We also present an algorithm for calculating the number of fuzzy cycles. In this thesis we also consider perturbed monomial systems over the field of p-adic numbers. This is done in Paper I. This systems are generated by polynomials (4.6) hq (x) = xn + q(x), 1 This

14

concept has nothing to do with the fuzzy set theory.

References

where the perturbation q(x) is a polynomial whose coefficients have a small p-adic absolute value. We investigate the connection between monomial and perturbated monomial systems by use of Hensel’s lemma. As in the monomial case the interesting dynamics of perturbated systems are essentially located on the unit sphere in Qp . Sufficient conditions on the perturbation for the two systems to have similar properties are derived. By similar properties we mean that there is a one to one correspondence between fixed points and cycles of the two kinds of systems.

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