Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods

TL;DR

This paper studies the behavior of random complex dynamical systems, showing that under certain conditions, they exhibit negative Lyapunov exponents and convergence to roots, revealing phenomena not seen in deterministic systems.

Contribution

It introduces new phenomena in random complex dynamics, including negativity of Lyapunov exponents and convergence to periodic measures, and applies these results to root-finding algorithms.

Findings

Negative Lyapunov exponents for generic systems

Convergence of measures to periodic cycles

Almost sure convergence of random orbits to polynomial roots

Abstract

We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere. We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values $z$ in the Riemann sphere, for almost every sequence of maps $\gamma =(\gamma_{1},\gamma_{2},\ldots )$, the Lyapunov exponent of $\gamma $ at $z$ is negative. Also, we show that for a generic system, for every initial value $z$ in the Riemann sphere, the orbit of the Dirac measure at $z$ under the iteration of the dual map of the transition operator tends to a periodic cycle of measures in the space of probability measures on the Riemann sphere. Note that these are new phenomena in random complex dynamics which cannot hold in deterministic complex dynamical systems. We apply the above theory and results of random complex dynamical systems to finding roots of any polynomials by random relaxed Newton's methods and we show that for any polynomial $g$, for any initial value $z$ in the complex plane which is not a root of $g'$, the random orbit starting with $z$ tends to a root of $g$ almost surely, which is the virtue of the effect of randomness.