  In this paper, we study two properties of the Lyapunov exponents under small
perturbations: one is when we can remove zero Lyapunov exponents and the other
is when we can distinguish all the Lyapunov exponents. The first result shows
that we can perturb all the zero integrated Lyapunov exponents $\int_M
\lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic
diffeomorphism. The second part contains an example which shows the local
genericity of diffeomorphisms with non-simple spectrum and three results: one
discusses the relation between simple-spectrum property and the existence of
complex eigenvalues; the other two describe the difference on the spectrum
between the diffeomorphisms far from homoclinic tangencies and those in the
interior of the complement. Moreover, among the conservative diffeomorphisms
far from tangencies, we prove that ergodic ones form a residual subset.
