Strengthened large deviations for rational maps and Gibbs fields, with unified proof

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 explicit 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 new large deviation principles for various nets of measures. We then specialize by introducing interactions and related physical quantities. This approach allows us to strengthen the classical large deviation results concerning Gibbs fields, in particular by allowing general van Hove nets. Landford results for finite range intercations are generalized in various ways to summable interactions.