Circle diffeomorphisms: rotation number and Poincaré-Denjoy theorem; structurally stable diffeomorphisms; Comments on global differentiable linearization. Hyperbolic fixed point and topological linearization. Stable variety theorem and slope lemma. Genericity of hyperbolic periodic orbits and transversal connections of saddles (Kupka-Smale theorem). Hyperbolic sets: stable and unstable foliations; examples: horseshoe, solenoid, Anosov derived diffeomorphism, Plykin attractor. Persistence and stability of hyperbolic sets; shading lemma. Stability of globally hyperbolic (Anosov) diffeomorphisms. Filtering and spectral decomposition of axiom A diffeomorphisms. Omega-stability theorem. Cycles and examples of omega-stable systems. Stability of saddle cross-links. Principle of reduction of dynamics to the central variety. Comments on the stability and omega-stability conjectures. Recurrences of vector fields on surfaces. Comments on the density of stable fields. Closing Lemma and related 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. Original version: 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.
* Basic syllabus. The teacher has the autonomy to make any changes.