Generic Bifurications for Involutary Area Preserving Maps.

Intro -- TABLE OF CONTENTS -- CHAPTER 1. INTRODUCTION -- CHAPTER 2. PRELIMINARIES -- 2.1 Terminology and Some Useful Results -- 2.1.1 Basic Ideas -- 2.1.2 The Weak and the Strong Topologies -- 2.1.3 Transversality -- 2.2 A Curve of Fixed Points -- CHAPTER 3. BIFURCATION FROM A SYMMETRIC FIXED POINT WITH MULTIPLIERS ± 1 -- 3.1 A Generating Function for a Family of Area Preserving Maps -- 3.2 The Involutory Property and the Generating Functions -- 3.3 Fixed Points Near a Symmetric Fixed Point with Multipliers 1 -- 3.4 Periodic Points Near a Symmetric Fixed Point with Multipliers -1 -- CHAPTER 4. CONDITIONS ON GENERATING FUNCTIONS OCCUR GENERICALLY -- 4.1 A Dense, Open Set of Generating Functions -- 4.2 Preliminary Perturbations -- 4.3 D[sub(3)] and D[sub(4)] are Dense and Open -- 4.4 D[sub(1)] is Dense and Open -- CHAPTER 5. BIFURCATION FROM A SYMMETRIC FIXED POINT u[sub(o)] OF φ[sub(e)][sub(o)] WITH MULTIPLIERS WHICH ARE n[sup(th)] PRIME ROOTS OF UNITY, FOR n ≥ 3 -- 5.1 Periodic Points Bifurcating from (u[sub(o)], e[sub(o)]) -- CHAPTER 6. NORMALISATION OF A FAMILY OF INVOLUTORY AREA PRESERVING MAPS -- 6.1 The Linear Transformation -- 6.2 Normalisation of φ up to Terms of Order n…1 -- 6.3 Polar Coordinates -- CHAPTER 7. GENERIC BIFURCATIONS -- 7.1 The Generic Result -- 7.2 Proof of Lemma 7.1.2 when the Multipliers of φ[sub(e)][sub(o)] at u[sub(o) are 1 -- 7.3 Perturbation when the Multipliers of φ[sub(e)][sub(o)] at u[sub(o)]are n[sup(th)] prime Roots of Unity for n ≥ 2 eo ° 123 -- CHAPTER 8. FAMILIES OF INVOLUTORY AREA PRESERVING MAPS AND SYMMETRIC HAMILTONIAN SYSTEMS -- 8.1 Symmetric Hamiltonian Systems -- 8.2 Existence of Families of Involutory Area Preserving Maps -- APPENDIX 1 PROOFS OF TWO RESULTS USED IN CHAPTER 4 -- A1.1Proof of Proposition 4.2.2(ii) -- A1.2 Proof of Lemma 4.2.4 -- REFERENCES.

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