Rigorous computer-assisted bounds on the period doubling renormalization fixed point and eigenfunctions in maps with critical point of degree 4

Abstract

We gain tight rigorous bounds on the renormalization fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearization of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.