On analytic perturbations of a family of Feigenbaum-like equations

We prove existence of solutions ( ϕ , λ ) of a family of Feigenbaum-like equations(0.1) ϕ ( x ) = 1 + ϵ λ ϕ ( ϕ ( λ x ) ) − ϵ x + τ ( x ) , where ϵ is a small real number and τ is analytic and small on some complex neighborhood of ( − 1 , 1 ) and real-valued on R . 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 − ϵ x + 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.