On limit functions and their natural boundaries in infinite compositions of entire functions

摘要

In this article, we consider infinite sequences {Φ_n} of entire functions in the complex plane C defined as compositions of the form Φ_n = f_n o f_(n-1) o ... o f_1, (A) where each fn, n = 1, 2, ..., is an entire function, and the limit functions Φ of such sequences. Under reasonable conditions on the sequence fn, and for the cases where Φ exists and is a nonconstant analytic function, one finds that the boundary of the domain where {Φ_n} converges to Φ is in fact the natural boundary of Φ, and that this boundary satisfies certain "expansion" properties when considered under the composition of the fn's. We also consider the case of constant limit functions Φ. In the final section we discuss the connection between the coefficients of a power series representation of a nonconstant limit Φ and the sequence {an} of a one parameter family of entire functions fn(z) = f (an,z), whose composition as in (1) converges in some domain to Φ.