On Analytic Perturbations of a Family of Feigenbaum-like Equations

الأصل · EN

We prove existence of solutions (ϕ,λ) of a family of of Feigenbaum-like equations family ϕ(x)=1+ λ ϕ(ϕ(λx)) - x +τ(x), where is a small real number and τ is analytic and small on some complex neighborhood of (-1,1) and real-valued on. The family (family) appears in the context of period-doubling renormalization for area-preserving maps (cf. GK). Our proof is a development of ideas of H. Epstein (cf Eps1, Eps2, Eps3) adopted to deal with some significant complications that arise from the presence of terms x +τ(x) in the equation (family). The method relies on a construction of novel 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.