Non-i.i.d. random holomorphic dynamical systems and the generic dichotomy

TL;DR

This paper studies non-i.i.d. random holomorphic dynamical systems with Markovian dependencies, revealing a generic dichotomy where systems are either mean stable with negative Lyapunov exponents or chaotic with full Julia sets.

Contribution

It establishes that generically such systems are either mean stable or chaotic, and proves that the set of mean stable systems has full measure in certain families, a new result even for i.i.d. cases.

Findings

Generically, systems are either mean stable or chaotic.

Mean stable systems have uniformly negative Lyapunov exponents.

The set of mean stable systems has full measure under certain conditions.

Abstract

We consider non-i.i.d. random holomorphic dynamical systems whose choice of maps depends on Markovian rules. We show that generically, such a system is mean stable or chaotic with full Julia set. If a system is mean stable, then the Lyapunov exponent is uniformly negative for every initial value and almost every random orbit. Moreover, we consider families of random holomorphic dynamical systems and show that the set of mean stable systems has full measure under certain conditions. The latter is a new result even for i.i.d. random dynamical systems.