Preprint A209/2003
Lyapunov exponents: How frequently are dynamical systems hyperbolic ?

Marcelo Viana | Bochi, Jairo

**Keywords: **
Lyapunov exponent | linear cocycle | non-uniform hyperbolicity

Lyapunov exponents measure the asymptotic behavior of tangent
vectors under iteration, positive exponents corresponding to
exponential growth and negative exponents corresponding to
exponential decay of the norm. Assuming {\em hyperbolicity,\/}
that is, that no Lyapunov exponents are zero, Pesin theory
provides detailed geometric information about the system, that is at the basis of several deep results on the dynamics of hyperbolic systems. Thus, the question in the title is central to the whole theory.
Here we survey and sketch the proofs of several recent results on genericity of vanishing and non-vanishing Lyapunov exponents. Genericity is meant in both topological and measure-theoretical sense. The results are for dynamical systems (diffeomorphisms) and for linear cocycles, a natural generalization of the tangent map which has an important role in Dynamics as well as in several other areas of Mathematics and its applications.
The first section contains statements and a detailed discussion of main results. Outlines of proofs follow. In the last section and the appendices we prove a few useful related results.