Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity

TL;DR

This paper studies non-i.i.d. random holomorphic dynamical systems on the Riemann sphere, focusing on the probability of tending to infinity and characterizing Julia sets under Markov chain-based randomness.

Contribution

It generalizes known results from i.i.d. systems to non-i.i.d. cases and provides conditions for the continuity of the probability function T and its relation to Julia sets.

Findings

Conditions for continuity of T on the entire space

Characterization of Julia sets via T under certain assumptions

Extension of classical results to Markov chain-based systems

Abstract

We consider random holomorphic dynamical systems on the Riemann sphere whose choices of maps are related to Markov chains. Our motivation is to generalize the facts which hold in i.i.d. random holomorphic dynamical systems. In particular, we focus on the function $T$ which represents the probability of tending to infinity. We show some sufficient conditions which make $T$ continuous on the whole space and we characterize the Julia sets in terms of the function $T$ under certain assumptions.