Convergence of a series whose terms are iterates of quadratic maps

Abstract

The functional equation $k(p(x)) + k(x)= x, \ p(x)=x^2 +c$, was used to find quadratic invariant curves of a planar mapping. The continuity of its solutions $k$ on an interval is tied to its series representation through $\sum_{i=0}^{\infty}(p^{(2i)}(x)- p^{(2i+1)}(x))$, where the terms contain iterates of $p$. The intervals of convergence of the series deserve much attention. Because of the presence of iteration, such maximal intervals are sometimes difficult to determine. In this paper we show how numerical computations using Maple V5.1 and the use of discriminants and resultants may assist such development.