On Analytic Perturbations of a Family of Feigenbaum-like Equations

TL;DR

This paper proves the existence of solutions for a family of Feigenbaum-like equations related to period-doubling in area-preserving maps, using novel bounds and extending Epstein's ideas.

Contribution

It develops a new method with tight a-priori bounds to establish solutions for perturbed Feigenbaum equations, advancing understanding of period-doubling phenomena.

Findings

Existence of solutions for the perturbed Feigenbaum-like equations.

New bounds on the scaling parameter λ.

A new proof of the Feigenbaum-Coullet-Tresser function existence.

Abstract

We prove existence of solutions $(\phi,\lambda)$ 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 $\eps$ is a small real number and $\tau$ 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 $\eps x +\tau(x)$ 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 $\lambda$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function.