  For any hyperbolic rational map and any net of Borel probability measures on
the space of Borel probability measures on the Julia set, we show that this net
satisfies a strong form of the large deviation principle with a rate function
given by the entropy map if and only if the large deviation and the pressure
functionals coincide. To each such principles corresponds a new expression for
the entropy of invariant measures. We give the explicit form of the rate
function of the corresponding large deviation principle in the real line for
the net of image measures obtained by evaluating the function $\log|T'|$. These
results are applied to various examples including those considered in the
literature where only upper bounds have been proved. The proof rests on some
entropy-approximation property (independent of the net of measures), which in a
suitable formulation, is nothing but the hypothesis involving exposed points in
Baldi's theorem. In particular, it works verbatim for general dynamical
systems. After stating the corresponding general version, as another example we
consider the multidimensional full shift for which the above property has been
recently proved, and we establish large deviation principles for nets of
measures analogous to those of the rational maps case.
