  We show some level-2 large deviation principles for rational maps satisfying
a strong form of non-uniform hyperbolicity, called "Topological
Collet-Eckmann". More precisely, we prove a large deviation principle for the
distribution of iterated preimages, periodic points, and Birkhoff averages. For
this purpose we show that each H{\"o}lder continuous potential admits a unique
equilibrium state, and that the pressure function can be characterized in terms
of iterated preimages, periodic points, and Birkhoff averages. Then we use a
variant of a general result of Kifer.
