我們將會透過兩個基本且重要的例子，有關圓上的微分同胚函數與可積漢米爾頓系統，闡明擾動理論中的KAM理論的中心想法如何運作。在過程中我們會見到小除數問題與可微性的損失這兩個問題是在何處產生與如何被克服的。

We will study two basic and important examples about circle diffeomorphisms and Hamiltonian systems, to clarify the central ideas of the celebrated Kolmogorov-Arnold-Moser Theory in perturbation theory. Two difficulties those pioneers encounter in early 20th century - problems of the small divisor and loss of differentiability, will be revealed and overcame in our procedures also.

1 Introduction 1
2 Circle Diffeomorphisms: The conjugated problems 3
2.1 Rotation number 3
2.2 Circle Diffeomorphisms 5
2.3 Arnold's Theorem 9
3 KAM Methodology for Arnold's Theorem 12
3.1 Analysis of Linearized Equation 12
3.2 The Newton Method in Banach Spaces 13
4 Analyic Diffeomorphism with Singular Conjugacy 20
4.1 Invariant measures and regularity of conjugacies 20
4.2 A Counterexample 22
5 KAM for Nearly Integrable Hamiltonian Systems 25
5.1 Formulation of the problem 25
5.2 Analysis of the Linearized Equation 29
5.3 The Newton Method in Banach Spaces 32
A The Proofs of Some Estimates in Section 5 36
Reference 41

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