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Lectures on the Morse complex for infinite dimensional Alberto Abbondandolo and Pietro Majer
 

Summary: Lectures on the Morse complex for infinite dimensional
manifolds
Alberto Abbondandolo and Pietro Majer
November 7, 2004
Contents
Introduction 2
1 A few facts from hyperbolic dynamics 2
1.1 Adapted norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Linear stable and unstable spaces of an asymptotically hyperbolic path . . . . . . . 3
1.3 Morse vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Local dynamics near a hyperbolic rest point . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Local stable and unstable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 The Grobman-Hartman linearization theorem . . . . . . . . . . . . . . . . . . . . . 10
1.7 Global stable and unstable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 The Morse complex in the case of finite Morse indices 16
2.1 The Palais-Smale condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 The Morse-Smale condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 The assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Forward compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Consequences of compactness and transversality . . . . . . . . . . . . . . . . . . . 19

  

Source: Abbondandolo, Alberto - Scuola Normale Superiore of Pisa

 

Collections: Mathematics