  We study skew-product dynamics for a large class of finitely-generated
semi--hyperbolic semigroups of rational maps acting on the Riemann sphere,
which generalizes both the theory of iteration of a single rational map of a
single complex variable complex/holomorphic dynamics) and the theory of
countable alphabet conformal iterated function systems (CIFSs). We construct
the thermodynamic formalism for such dynamical systems and geometric potentials
by developing the notion of nice families that extend to the case of our highly
disconnected skew product phase space the powerful notion of nice sets due to
Rivera--Letelier and Przytycki, and the allied earlier notion of $K(V)$ sets
due to Denker and the last named author. We leverage out techniques to prove
the existence and uniqueness of equilibrium states for a wide class of H\"older
potentials, and concomitant statistical laws: central limit theorem, law of
iterated logarithm, and exponential decay of correlations. We devote lots of
space and effort to control (non-recurrent) critical points which is a
notoriously challenging task even for a single rational function; more
generators add qualitatively new challenges. Beyond dynamics, but still with
dynamical methods, we advance the study of finer fractal geometrical properties
of the intricate Julia sets associated to such systems and, in particular, via
equilibrium states, we perform a multifractal analysis of Lyapunov exponents.
We use the Nice Open Set Condition (NOSC) introduced by the last two authors,
and apply our new techniques to settle a long-standing problem in the theory of
rational semigroups by proving that for our class of semigroups the Hausdorff
dimension of each fiber Julia set is strictly smaller than the Hausdorff
dimension of the global Julia set of the semigroup.
