On analytic perturbations of a family of Feigenbaum-like equations

We prove existence of solutions (ϕ,λ)(ϕ,λ) of a family of Feigenbaum-like equationsequation(0.1)ϕ(x)=1+ϵλϕ(ϕ(λx))−ϵx+τ(x), where ϵ is a small real number and τ   is analytic and small on some complex neighborhood of (−1,1)(−1,1) and real-valued on RR. The family (0.1) appears in the context of period-doubling renormalization for area-preserving maps (cf. Gaidashev and Koch (preprint) [7]). Our proof is a development of ideas of H. Epstein (cf. Epstein (1986) [2], Epstein (1988) [3], Epstein (1989) [4]) adopted to deal with some significant complications that arise from the presence of the terms −ϵx+τ(x)−ϵx+τ(x) in Eq. (0.1). 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.