A diffeomorphism ƒ : [0, 1] → [0, 1] is called Morse-Smale if f has only hyperbolic periodic points. 1. Prove that a Morse-Smale diffeomorphism has only finitely many periodic points. 2. Prove that a Morse-Smale diffeomorphism of [0, 1] is structurally stable. 3. Prove that the map ƒ(r) = x³ + ³r is a Morse-Smale diffeomorphism on the interval [-1/2,1/2].
A diffeomorphism ƒ : [0, 1] → [0, 1] is called Morse-Smale if f has only hyperbolic periodic points. 1. Prove that a Morse-Smale diffeomorphism has only finitely many periodic points. 2. Prove that a Morse-Smale diffeomorphism of [0, 1] is structurally stable. 3. Prove that the map ƒ(r) = x³ + ³r is a Morse-Smale diffeomorphism on the interval [-1/2,1/2].
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 16CM
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