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.

25 pages. Published in Nonlinearity 35 (2022) 1857--1875. The published version of this paper contains serious typos in Main result C (we asked the publisher of the journal to fix the typo several times before the publication of this paper, but it did not work), which have been corrected in the arXiv version

References in corpus (3)

• Pointwise Hölder Exponents of the Complex Analogues of the Takagi Function in Random Complex Dynamics
• Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods
• Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity

Cited by in corpus (1)

• On the stochastic bifurcations regarding random iterations of polynomials of the form $z^2+c_n$