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

## TL;DR

This paper studies the distribution of pre-images and periodic points for a skew-product map generated by finitely many rational maps, analyzing ergodic properties and orbit deviations in complex dynamics.

## Contribution

It introduces new results on the distribution of points and ergodic measures for skew-product maps driven by rational functions with varying degrees.

## Key findings

- Distribution of pre-images and periodic points characterized
- Ergodicity of equilibrium measures analyzed
- Orbit deviation estimates provided

## Abstract

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.