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

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