臺灣博碩士論文加值系統

本文最要是把Poincare-Birkhoff fixed point theorem的證明清楚的寫下來,因為當 初Birkoff給的證明還有一些地方不是很清楚,再加上這個定理在撞球周期軌跡問 題上一個有趣的應用。

This paper further modifies Brown and Neumann’s proof to make it as clear as possible, clarifying what the previous renditions have left unclear.Moreover, the periodic trajectories problem of billiard is chosen as a application of the Poincare-Birkhoff fixed point theorem.

誌謝 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
中文摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
1 Introduction 1
2 Definitions and Statment of The Theorem 1
3 Rotation Numbers and Index 2
4 Proof of The Theorem 4
5 Application to Billiard Ball Problem 11
6 Final Remarks 14
References 15

[1] G. D. Birkhoff, Proof of Poincare’s geometric theorem. Trans. Amer. Math. Soc.14 (1913), 14-22
[2] G. D. Birkhoff, An extension of Poincare’s last geometric theorem. Acta. Math. 47 (1925), 297-311
[3] G. D. Birkhoff, Dynamical system, Amer, Math. Soc., Providence, RI, 1966
[6] H. Poincarex, Sur un theoreme de geometrie. Rend. Circ. Mat. Palermo 33 (1912), 375-407
[4] M. Brown and W. D. Neumann, Proof of the Poincare’-Birkhoff fixed point theroem. Michigan Math. J 24, Issume 1 (1997), 375-407
[5] M. Morse, GeorgeDavid Birkhoffand his mathematical work, Bull. Amer. Math. Soc. 52 (1946), 357-391.
[7] A. Silva, Lectures on Symplectic Geometry, Springer-Verlag, Lecture Notes in Math- ematics, 2001,
[8] S. Tabachnikov, Geometry and Billiards, Amer. Math. Soc., 2005