關於一維區間函數的一些動態性質
研究生 : 劉孔驪 指導教授 : 莊 重 博士
國立交通大學
應用數學系
摘 要
此論文包含三章。第一章的主題是針對單峰函數 來討論此一集合{x belongs to I:S(x)=K(f)} 。首先，我們考慮各種可能的狀態。而達到我們目的的主要手法，是來自於定理1.2.10的證明中所介紹的證明技巧。其次，我們專注於二次函數族 fr(x)=rx(1-x) (quadratic family)，使用了重新正規化的技巧(renormalization techniques)，導出當二次函數族在歷經倍週期分歧(period-doubling bifurcation)的過程時，其所對應之集合 {x belongs to I:S(x)=K(fr)}的精確形式為何。
在第二章中，我們專注在檢驗書本[1]之1.11及1.19節裡，關於“許瓦爾茲迅導數小於零(Schwarzian derivative of Sf < 0) ”的一些推論是否仍然成立，倘若將其替代為“ 在區間 中有敏感度( f has sensitive dependence on initial data on I )”的話。在最後一章裡，我們首先將推廣1.18([1])節中的定義並使用米勒和舍斯頓(Milnor and Thurston)([4])所創造的符號。除此之外，我們也發現了確保分段單調函數(piecewise-monotone map)的符號序列(symbolic sequence)是許可的(admissible)充份條件。最後，有關分段單調函數之週期點所成集合的結果聲明與斷言亦包過括於本章中。

On Some Dynamical Properties of Interval Maps
Student: Kong-Li Liu Adviser: Jonq Juang
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 30050, Taiwan, R.O.C.
Abstract
The dissertation contains three chapters. The topic of chapter 1 is to discuss the set {x belongs to I:S(x)=K(f)} for a unimodal map. First, we consider all possible scenarios. The main techniques to achieve so are those introduced in the proof of Theorem 1.2.10. Let fr(x)=rx(1-x) be the quadratic map. We then use renormalization techniques to derive the precise form of the set {x belongs to I:S(x)=K(fr)} as the family of fr undergoes period doubling bifurcation.
In chapter 2, we devote our attention to checking whether some conclusions concerning the assumption " Sf<0 on I " in §1.11, §1.19([1]) still hold if we substitute it for "sensitivity on initial data on I." In the third chapter, we will first provide the generalization of definitions in §1.18([1]) and use the notations developed by J. Milnor and W. Thurston([4]) throughout this chapter. Moreover, we derive a sufficient condition to guarantee the admissibility of symbolic sequences associated with piecewise-monotone maps(i.e. l-modal maps, l>1 ). Additionally, some assertions about the set of periodic points of such maps will be included.

Contents
Acknowledgements i
Contents ii
List of Figures iii
Chapter 1 The Set {x belongs to I:S(x)=K(f)} 1
1.1 Introduction 1
1.2 Preliminaries 2
1.3 Main Results (Part A) 8
1.4 Main Results (Part B) 22
1.5 Conclusions 36
Chapter 2 Replace “ Sf<0 ” by “Sensitivity” 38
2.1 Introduction 38
2.2 Preliminaries 39
2.3 Main Results (Part A) 40
2.4 Main Results (Part B) 45
2.5 Main Results (Part C) 46
2.6 Conclusions 48
Chapter 3 Piecewise-Monotone Maps 50
3.1 Introduction 50
3.2 Preliminaries 51
3.3 Main Results 53
3.4 Conclusions 61
Bibliography 62

Bibliography
[1] R. Devaney, An Introduction To Chaotic Dynamical Systems, Addison-Wesley, New York, 1989.
[2] G. Chen, T. Huang, Chaotic behavior of interval maps as characterized by unbounded growth of total variations of their n-th iterates as n →∞, submitted.
[3] J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys. 70, 133-160(1979)
[4] J. Milnor, C. Tresser, On entropy and monotonicity for real cubic maps. Commun. Math. Phys. 209, 123-178(2000)
[5] W. de Melo, S. van Strien, One-Dimensional Dynamics. Springer-Verlag, N.Y.,1991.
[6] J. Milnor, W. Thurston, On iterated maps of the interval. I,II. Preprint Princeton. Published in: “Dynamical systems: Proc. Univ. of Maryland 1986-87”, Lecture Notes in Math. 1342, Springer, Berlin New York, 465-563(1977)
[7] W. Parry, Symbolic dynamics and transformations of the unit interval. Trans. A.M.S. 122, 368-378(1966)
[8] M. Metropolis, M.L. Stein and P.R. Stein, On finite limit sets for transformations of the unit interval. J. Combin. Theory 15, 25-44(1973)