Hyberbolic Dynamics

Diffeomorphisms of the circle: rotation number and Poincaré-Denjoy theorem; structurally stable diffeomorphisms; Comments on global differentiable linearization. Hyperbolic fixed point and topological linearization. Theorem of the stable manifold and the inclination lemma. Genericity of periodic hyperbolic orbits and transversal saddle connections (Kupka-Smale theorem). Hyperbolic sets: stable and unstable foliations; example: horseshoe, solenoid, Anosov derived diffeomorphism, Plykin’s attractor. Persistence and stability of hyperbolic sets; shadowing lemma. Stability of globally hyperbolic diffeomorphisms Anosov). Filtration and spectral decomposition of Axiom-A diffeomorphisms. Theorem of Ω-stability. Cycles and examples of Ω -unstable systems. Stability of transversal saddle connection. Principle of dynamic reduction to the central manifold. Comments on conjectures on stability and on Ω-stability. Recurrence of vector fields on surfaces. Comments on density of stable fields. Closing lemma and correlated issues. Elements of bifurcation theory.

References:
MELO, W., VAN STRIEN, S. – One-Dimensional Dynamics, Springer-Verlag, 1993.
PALIS, J., DE MELO, W. – Introduction to Dynamical Systems, Berlin, Springer-Verlag, 1982. Versão Original: Projeto Euclides, IMPA, 1987.
PALIS, J., TAKENS, F. – Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations, Cambridge University Press, 1993.
SHUB, M. – Global Stability of Dynamical Systems. New York, Springer-Verlag, 1987.

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