On Analytic Perturbations of a Family of Feigenbaum-like Equations

We prove existence of solutions $(\varphi ,\lambda )$ of a family of of Feigenbaum-like equations \label{family} \phi(x)={1+\eps \over \lambda} \phi(\phi(\lambda x)) -\eps x +\tau(x), where $\text{eps}$ is a small real number and $\tau$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\text{fR}$. The family $(\text{ref}family)$ 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 $\text{eps}x+\tau (x)$ in the equation $(\text{ref}family)$. 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 $\lambda$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function.