  We show how the small perturbations of a linear cocycle have a relative
rotation number associated with an invariant measure of the base dynamics an
with a $2$-dimensional bundle of the finest dominated splitting (provided that
some orientation is preserved). Likewise small perturbations of diffeomorphisms
have a relative rotation number associated with an invariant measure supported
in a hyperbolic set and with a $2$-dimensional bundle as above.
  The properties of that relative rotation number allow some steps towards
dichotomies between complex eigenvalues and dominated splittings in higher
dimensions and higher regularity. We also prove that generic smooth linear
cocycles above a full-shift (and actually above infinite factors of transitive
subshift of finite type) admit a periodic point with simple Lyapunov spectrum.
