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

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.

On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$

“…By using the hyperbolic metric, we have the following characterization of attracting minimal sets. For the proof, see [27,Remark 3.5] or [30,Lemma 2.8].…”

“…Compared with the autonomous case, the following results show that we can solve random versions of the abovementioned conjectures. For the proof, see [30] and [27].…”

“…are complexvalued independent random variables that follow the uniform distribution on the closed disk B(c, r) = {c ′ ∈ C : |c ′ − c| ≤ r} on the c-plane. The reader is referred to Remark 4.10 of [30]. See Setting 4.1 for the rigorous setting.…”

“…For details, see [27,28]. See also [29,30] for the author generalizing the setting from i.i.d. to non-i.i.d.…”

On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$

“…are complex-valued independent random variables that follow the uniform distribution on the closed disk B(c, r) = {c ∈ C : |c − c| ≤ r} on the c-plane. The reader is referred to [31,Remark 4.10]. See Setting 4.1 for the rigorous setting.…”