On Analytic Perturbations of a Family of Feigenbaum-like Equations

Abstract

We prove existence of solutions $(ϕ,λ)$ of a family of of Feigenbaum-like equations \label{family} ϕ(x)={1+\eps \over λ} ϕ(ϕ(λx)) -\eps x +τ(x), where $\eps$ is a small real number and $τ$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\fR$. The family $(\ref{family})$ appears in the context of period-doubling renormalization for area-preserving maps (cf. \cite{GK}). Our proof is a development of ideas of H. Epstein (cf \cite{Eps1}, \cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms $\eps x +τ(x)$ in the equation $(\ref{family})$. The method relies on a construction of novel {\it a-priori} bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter $λ$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function.