  We consider the thermodynamic formalism of a complex rational map $f$ of
degree at least two, viewed as a dynamical system acting on the Riemann sphere.
More precisely, for a real parameter $t$ we study the (non-)existence of
equilibrium states of $f$ for the potential $-t \ln |f'|$, and the analytic
dependence on $t$ of the corresponding pressure function. We give a fairly
complete description of the thermodynamic formalism of a rational map that is
"expanding away from critical points" and that has arbitrarily small "nice
sets" with some additional properties. Our results apply in particular to
non-renormalizable polynomials without indifferent periodic points, infinitely
renormalizable quadratic polynomials with a priori bounds, real quadratic
polynomials, topological Collet-Eckmann rational maps, and to backward
contracting rational maps. As an application, for these maps we describe the
dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the
measure of maximal entropy, and obtain some level-1 large deviations results.
