  We study the dimension spectrum for Lyapunov exponents for rational maps
acting on the Riemann sphere and characterize it by means of the
Legendre-Fenchel transform of the hidden variational pressure. This pressure is
defined by means of the variational principle with respect to non-atomic
invariant probability measures and is associated to certain $\sigma$-finite
conformal measures. This allows to extend previous results to exceptional
rational maps.
