  In this paper we revisit uniformly hyperbolic basic sets and the domination
of Oseledets splittings at periodic points. We prove that periodic points with
simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual
diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the
C1-topology we can prove that either all periodic points of a hyperbolic basic
piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f
has a finest dominated splitting into one-dimensional sub-bundles and all
Lyapunov exponent functions of f are continuous in the weak*-topology) or it
can be C1-approximated by an equidimensional cycle associated to periodic
points with robust different signatures. The later can be used as a mechanism
to guarantee the coexistence of infinitely many periodic points with different
signatures.
