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研究生： 李亭芳
Ting-Fang Li
論文名稱：
The average of the number of r-periodic points over a quadratic number field.
指導教授： 夏良忠
Liang-Chung Hsia
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 94
語文別： 英文
論文頁數： 39
中文關鍵詞： 動態系統
外文關鍵詞： p-adic
相關次數： 點閱：15  下載：0 
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    • 在這篇論文中，我們要計算在一個二次的field extension上週期為 的點個數的平均值，其構想和方法主要是參考 [3] 和 [4] 這兩篇論文。我們利用兩種不同的方法去計算這一個平均值，Prime Number Theorem 和Group Action。第一個方法是先計算週期為 的點個數，再利用Prime Number Theorem去計算平均值。第二個方法是去討論這個平均值和Galois group 作用在這些點上的orbit個數間的關係，進而利用這樣的關係計算出此平均值。

    In this paper, we compute the average of the number of r-periodic points
    over a quadratic number ¯eld generalizing results in [3] and [4]. We use two
    di®erent methods, the prime number theorem and group action, to compute
    the average and compare the result. First method is to counte the number of
    the primitive r-periodic points. After that we use the prime number theorem
    to compute the average. And we discuss relationship between the average and
    the number of orbits in the set of primitive r-periodic points under the Galois
    action in the second method.

    Contents 1 Introduction 1 2 Counting the number of periodic points 2 3 The prime number Theory 6 4 limt!11¼K(t)Xp·tPr(h;K}) 7 4.1 K = Q(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 K = Q(w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 K = Q(sqrt{q}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 ChebotarÄev''s Density Theorem 29 6 Group Action 30 7 limt!11¼K(t)Xp·tPr(h;K}) again 34 Reference 39

    References
    [1] Jean-Pierre Serre, On a Thoerem of Jordan, American Mathematical Society. V. 40,N 4.
    [2] Neal Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-functions, Springer-Verlag.
    [3] M. Nilsson, Monomial Dynamics in the Finite Field Extensions of the Fields of p-adic Numbers, London Mathematical Society.
    [4] M. Nilsson and R. Nyqvist, The Asymptotic Number of Periodic Points of Discrete p-adic Dynamical Systems, Tr. Mat. Inst. Steklova 245 (2004), Izbr. Vopr. p-adich. Mat. Fiz. i Anal.; translation in Proc. Steklov Inst. Math. 2004,
    [5] K. Chandrasekharan, An Intoduction to the Analytic Number Theory,Springer-
    Verlag New York Inc, 1968.
    [6] Serge Lang, Algebra, 3rd ed. Springer-Verlag.
    [7] Serge Lang, Algebraic Number Theory, Springer-Verlag.
    [8] Alain M. Robert, A course in p-adic Analysis, New York Springer-Verlag, 2000.
    [9] Ireland Kenneth F. and Michael Rosen., A Classical Introduction to Modern Number Theory, 2nd edition, New York Springer-Verlag, 1982.
    [10] P. Morton and P. Patel, The Galois theory of periodic points of polynomial maps, Proc. London Math. Soc. 68 (1994), 224-263.
    [11] P. Stevenhagen and H.W. Jr. Lenstra, ChebotarÄev and his density Theorem, Math. Intell. 1996. V. 18, N 2.
    [12] W. J. le Veque, Topics in Number Theory, Addison-Wesley Publishing co., Reading Mass., 1956. 39

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