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.
Categories: math.DS