It is a theorem of Denker and Urbański that if T:ℂ→ℂ is a rational map of degree at least two and if ϕ:ℂ→ℝ is Hölder continuous and satisfies the “thermodynamic expanding” condition P(T,ϕ) > sup(ϕ), then there exists exactly one equilibrium state μ for T and ϕ, and furthermore (ℂ,T,μ) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on ℂ, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogenschütz. Specifically, if (T,Ω,P,θ) is a holomorphic random dynamical system on ℂ and ϕ:Ω→ ℋα(ℂ) is a Hölder continuous random potential function satisfying one of several sets of technical but reasonable hypotheses, then there exists a unique equilibrium state of (X,P,ϕ) over (Ω,Ρ,θ).

The lattice point problem in dynamical systems investigates the distribution of certain objects with some length property in the space that the dynamics is defined. This problem in different contexts can be interpreted differently. In the context of symbolic dynamical systems, we are trying to investigate the growth of N(T), the number of finite words subject to a specific ergodic length T, as T tends to infinity. This problem has been investigated by Pollicott and Urbański to a great extent. We try to investigate it further, by relaxing a condition in the context of deterministic dynamical systems. Moreover, we investigate this problem in the context of random dynamical systems. The method for us is considering the Fourier-Stieltjes transform of N(T) and expressing it via a Poincaré series for which the spectral gap property of the transfer operator, enables us to apply some appropriate Tauberian theorems to understand asymptotic growth of N(T). For counting in the random dynamics, we use some results from probability theory.