Generic Symplectic Cocycles are Hyperbolic

Michel Cambrainha de Paula

**Keywords:**symplectic | Linear cocycles | pinching | twisting

The goal of this thesis is to give a positive and complete answer to the question about genericity of nonuniformly hyperbolic dynamics for two classes of symplectic cocycles: the locally constant ones and the Holder continuous and dominated ones, both over shifts. Indeed, we prove that the Avila-Bonatti-Viana criterion for simplicity of Lyapunov spectra holds generically in these two settings. Moreover, to symplectic cocycles, simplicity implies (nonuniform) hyperbolicity, since the multiplicity of zero as a Lyapunov exponent must be even.