### 博碩士論文 93221023 詳細資訊

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(The average of the number of r-periodic points over a quadratic number field.)

 ★ 數論在密碼學上的應用 ★ a^n-b^n的原質因子，其中a,b為高斯整數 ★ Group Representations on GL(2,F_q) ★ Legendre的定理在Z[i]和Z[w]的情形 ★ Diophantine approximation and the Markoff chain ★ 週期為r之週期點個數的平均值 ★ 橢圓曲線上扭點的平均數 ★ 正特徵值函數體上的逼近指數之研究 ★ On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic ★ ZCm 的理想環生成元個數之上限

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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.

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

[1] Jean-Pierre Serre, On a Thoerem of Jordan, American Mathematical Society. V. 40,N 4.
[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