  In this paper, we consider the dynamics of a skew-product map defined on the
Cartesian product of the symbolic one-sided shift space on $N$ symbols and the
complex sphere where we allow $N$ rational maps, $R_{1}, R_{2}, \cdots, R_{N}$,
each with degree $d_{i};\ 1 \le i \le N$ and with at least one $R_{i}$ in the
collection whose degree is at least $2$. We obtain results regarding the
distribution of pre-images of points and the periodic points in a subset of the
product space (where the skew-product map does not behave normally). We further
explore the ergodicity of the Sumi-Urbanskii (equilibrium) measure associated
to some real-valued H\"{o}lder continuous function defined on the Julia set of
the skew-product map and obtain estimates on the mean deviation of the
behaviour of typical orbits, violating such ergodic necessities.
