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研究生:歐迪興
研究生(外文):Dyi-Shing Ou
論文名稱:一維函數之共軛性
論文名稱(外文):Conjugacy of One Dimensional Maps
指導教授:彭栢堅
指導教授(外文):Ken Palmer
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2009
畢業學年度:97
語文別:英文
論文頁數:57
中文關鍵詞:動態系統拓樸共軛片段線性函數數值計算保測變換
外文關鍵詞:Dynamical SystemsTopological ConjugacyPiecewise Linear MapsNumerical CalculationMeasure Preserving Transformation
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  • 被引用被引用:0
  • 點閱點閱:339
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若一連續函數 f:[0,1]->[0,1] ,存在一組分割 0=z_0<z_1<...<z_n=1 使 f(z_2i)=0 、 f(z_2i+1)=1 且 f 在 [z_i,z_i+1] 區間為單調函數,則稱 f 為 n-modal。Milnor 及 Thurston (1977) 最先給出了一個片段絕對單調函數至片段線性函數之 semi-conjugacy 的存在性。本篇論文為推廣 Fotiades, Boudourides (2001) 及 Banks, Dragan, Jones (2003) 的方法,建構 n-modal 函數到 tent map 之 semi-conjugacy ,並更進一步利用此方法證出 semi-conjugacy 的唯一性。此方法可用於數值計算 n-modal 映射之 semi-conjugacy ,並詳細估計出其收歛性。由於前述 Fotiades 及 Banks 等人只給了當 conjugacy 存在的結果,本文給出所構造出的 semi-conjugacy 為一對一映成函數之等價條件,這些條件驗證了 Parry (1966) 的結果。本文最後給了兩個應用:一個是研究 logistic map l_mu(x)=mu x(1-x) 之 invariant Cantor set 隨 mu>=4 變化之軌跡,另一個是可建構 n-modal map 之保測變換。
A continuous map f:[0,1]->[0,1] is called an n-modal map if there is a partition P={0=z_0<z_1<...<z_n=1} such that f(z_2i)=0, f(z_2i+1)=1 and, f is monotone on each [z_i,z_i+1]. It was proved by Milnor and Thurston (1977) that there exists a topological semi-conjugacy from a piecewise strictly monotone map to a piecewise linear map. In this article, we give a method for constructing the topological semi-conjugacy numerically which extends the results from Fotiades, Boudourides (2001) and Banks, Dragan, Jones (2003). In addition, the uniqueness of the semi-conjugacy, is proved by this method. The convergence rate is discussed for the approximation method also. Moreover, in contrast to Fotiades and Banks who only consider condition which ensure the conjugacy map exists, here we state equivalent conditions for the semi-conjugacy to be exactly a bijection, which coincide with Parry''s (1966) result. Finally, two applications are given. In one, we study the trajectory of the invariant Cantor set for the logistic map l_mu(x)=mu x(1-x) when the parameter mu>=4. In the other, we construct an invariant measure for an n-modal map.
口試委員會審定書 ii
序 iii
中文摘要 iv
英文摘要 iv
1. Introduction 1
2. Definitions 2
3. The construction of the semi-conjugacy for n-modal maps 5
4. Examples of constructing conjugacy maps 15
5. The error analysis for the approximate conjugacy 19
6. Equivalent conditions for the existence of a conjugacy 24
7. Some analytic conjugacy solutions - the Chebyshev polynomials 28
8. Extensions 30
9. Invariant Measure 34
參考文獻 37
A. 附錄 38
A.1. The library for generating the approximation semi-conjugacy 38
A.2. The main program for generating the approximate semi-conjugacy 44
A.3. The main program for drawing the invariant cantor set 49
A.4. The configuration file 52
A.5. The Makefile for generationg the executable programs 53
A.6. Instructions for generating and running the programs 54
索引 57
J. Banks, V. Dragan, and A. Jones, Chaos: A mathematical introduction. Cambridge
University Press, 2003.
R. L. Burden and J. D. Faires, Numerical Analysis, 7th ed. Brooks/Cole, 2001.
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems.
Birkhauser, 1980.
Y.-C. Chen, “Family of invariant Cantor sets as orbits of differential equations,” International
Journal of Bifurcation and Chaos, vol. 18, no. 7, pp. 1825–1843, 2008.
G. H. Choe, Computational Ergodic Theory. Springer-Verlag, 2005.
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed. Addison-
Wesley, 1989.
N. A. Fotiades and M. A. Boudourides, “Topological conjugacies of piecewise monotone
interval maps,” International Journal of Mathematics and Mathematical Sciences,
vol. 25, pp. 119–127, 2001.
IEEE Standard for Floating-Point Arithmetic. The Institute of Electrical and
Electronics Engineers, Inc., 2008. [Online]. Available: http://ieeexplore.ieee.org/xpl/
freeabs_all.jsp?arnumber=4610935
A. S. Kechris, Classical Descriptive Set Theory. Springer-Verlag, 1995.
J. Milnor andW. Thurston, Dynamical Systems : proceedings of the special year held at
the University of Maryland, College Park, 1986-87, ser. Lecture Notes in Mathematics.
Springer-Verlag, 1988, vol. 1342, ch. On iterated maps of the interval, pp. 465–563.
W. Parry, “Symbolic dynamics and transformations of the unit interval,” Transactions of
the American Mathematical Society, vol. 122, pp. 368–378, 1966.
H. L. Royden, Real Analysis, 3rd ed. Prentice Hall, 1988.
R. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis.
Marcel Dekker, 1977.
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