Holomorphic solutions to pantograph type equations with neutral fixed points

Pantograph type equations have been studied extensively owing to the numerous applications in which these equations arise. These studies focused primarily on the case when the functional argument is linear, and the origin is either a repelling or attracting fixed point. The nonlinear case has been studied by Oberg [Trans. Amer. Math. Soc. 161 (1971) 302–327] and Marshall et al. [J. Math. Anal. Appl. 268 (2002) 157–170], but the focus again was on repelling or attracting fixed points. Oberg (op. cit.), however, did consider briefly the neutral fixed point case and found a connexion with Siegel discs. In this paper we build on Oberg’s work and study the neutral fixed point case. We show that, for nonlinear functional arguments with neutral fixed points, pantograph type equations have nonconstant holomorphic solutions only if the functional argument has a Siegel disc centered at the fixed point. We then show that the boundary of the Siegel disc forms a natural boundary for the nonconstant holomorphic solutions.  2004 Elsevier Inc. All rights reserved.