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Holomorphic solutions to linear first-order functional differential equations. (English) Zbl 1101.34049

Summary: We study holomorphic solutions to linear first-order functional-differential equations that have a nonlinear functional argument. We focus on the existence of local solutions at a fixed-point of the functional argument and the holomorphic continuation of these solutions. We show that the Julia set for the functional argument dominates not only the conditions for holomorphic continuation, but also the existence of local solutions. In particular, nonconstant holomorphic solutions in a neighbourhood of a repelling or neutral fixed-point are uncommon in that the functional argument must satisfy conditions that force it to have an exceptional point in the former case, and a Siegel fixed-point in the latter case. In contrast, local holomorphic solutions always exist near attracting fixed-points. In this case, a subset of the Julia set forms a natural boundary for holomorphic continuation.

MSC:

34K05 General theory of functional-differential equations
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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