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Monomial Dynamical Systems in the Fields of P-Adic Numbers and Their Finite Extensions

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Monomial Dynamical Systems in the Fields of P-Adic Numbers and Their Finite Extensions 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 n by iterations of h(x)=x ,wheren 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)inKp will vary very much with p. We find explicit formulas for Pr(h(x), Kp), by using methods from number theory, for example M¨obius inversion. One of the main parts of this thesis is the computation of the limit 1 lim Pr(h, Kp), t→∞ π(t) pt 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)inKp, 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 cy- cles, 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 in- spiring ideas and advices. I also want to thank my co-supervisor Anders Melin, University of Lund. Scientific contacts with him were very impor- tant 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´ed’Auvergnefortheir 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 1 Introduction 3 2 Fields of p-adic Numbers 4 2.1Non-Archimedeanfields..................... 4 2.2 The field of p-adicnumbers................... 5 2.3 Extensions of the field of p-adicnumbers............ 7 2.4Hensel’slemma.......................... 9 2.5Rootsofunity........................... 9 3 Discrete Dynamical Systems 11 3.1Periodicpointsandtheircharacter............... 11 4 Summary of the Thesis 13 References 15 I Cycles of Monomial and Perturbed Monomial p-adic Dy- namical Systems 19 1 Introduction 21 2 Properties of Monomial Systems 21 3NumberofCycles 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 Sys- tems 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 2RootsofUnity 61 2.1 Roots of unity in Fp ....................... 62 2.2Cyclotomicpolynomials..................... 63 3 Periodic Points and Cyclotomic Polynomials 64 4 Group Action on Finite Sets 67 5 Periodic Points in Finite Fields 68 vii 6UptoQp 69 6.1 The p-adicfields......................... 70 6.2 Roots of unity in Qp ....................... 71 6.3 Periodic points in p-adicfields.................. 72 References 73 IV Fuzzy Cycles of p-adic Monomial Dynamical Systems 75 1 Introduction 77 2 Global Dynamics 79 3 Local Dynamics 82 3.1Dynamicsaroundneutralpoints................ 85 3.2Dynamicsaroundattractors................... 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 3RootsofUnity 98 4 Monomial Dynamics 100 5 Counting Periodic Points 101 6 Asymptotic Behavior 103 7 Periodicity 106 References 109 viii General Introduction 1 Introduction 1 Introduction Almost everything in nature and society that evolves in time can be de- scribed 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 dis- crete. A continuous dynamical system describes phenomena that evolve continuously in time, while a discrete dynamical system describes phenom- ena 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 measur- able 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.Themetricρ 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.Thep-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.
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