On Analytic Perturbations of a Family of Feigenbaum-like Equations

We consider a family of of Feigenbaum-like equations φ(y) = 1 + ǫ λ φ(φ(λy))− ǫy + τ(y), (0.1) where ǫ is a real number and τ is analytic on some complex neighborhood of (−1, 1) and real-valued on R. We first give a motivation for studying functional equations of this kind. Specifically, we give a heuristic argument that, in a space of two-dimensional maps, the “universal” area-preserving map which has all orbits of period 2, k ∈ Z, is very close to a Henon-like map of the form H(x, u) = (φ1,0(x)− u, x− φ1,0(φ1,0(x)− u)), where φǫ,τ is a solution of (0.1). The problem of existence of solutions of (0.1) for ǫ ≈ 1 turns out to be rather difficult. In this paper we describe an approach to a more accessible case of small ǫ and all small analytic τ whose derivative is also sufficiently small. Our proof of existence of solutions φǫ,τ is analytic in nature but uses numerical computations and rigorous computer-assistance to verify some technical conditions. The approach is a development of ideas of H. Epstein [Eps1] and D. Sullivan [Sul] adopted to deal with some significant complications that arise from the presence of terms ǫy + τ(y) in the equation (0.1). The method relies on a construction of novel a-priori bounds for the function φτ,ǫ 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. Analytic Perturbations of Feigenbaum-like Equations 1