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In this paper, we consider the dynamics of a skew-product map defined on theCartesian product of the symbolic one-sided shift space on $N$ symbols and thecomplex 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 thecollection whose degree is at least $2$. We obtain results regarding thedistribution of pre-images of points and the periodic points in a subset of theproduct space (where the skew-product map does not behave normally). We furtherexplore the ergodicity of the Sumi-Urbanskii (equilibrium) measure associatedto some real-valued H\"{o}lder continuous function defined on the Julia set ofthe skew-product map and obtain estimates on the mean deviation of thebehaviour of typical orbits, violating such ergodic necessities.

Distribution of typical orbits for a skew-product map generated by random dynamics of finitely many rational maps