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

Abstract

We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point. We prove that the fixed point is hyperbolic and use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation and for the scaling of additive noise. We find analytic extensions of the fixed point function to larger domains. We use multi-precision arithmetic with rigorous directed rounding to bound operations in a space of analytic functions yielding tight bounds on power series and universal constants.

Rigorous computer-assisted bounds on renormalisation fixed point functions, eigenfunctions, and universal constants

“…In this paper we follow the method established in our paper [16] where we studied the universality class corresponding to degree 4 critical points. In section 2, we work with a modified renormalisation operator, on a suitable space of analytic functions, corresponding to the action of the usual doubling operator on even maps.…”

“…Secondly, we bound the action of DΦ(G) (for all G ∈ B) on a single ball B H of high-order functions f H , such that f H ≤ 1, that therefore contains all of the high-order basis elements e k for k > N . The latter requires careful consideration of the action of DΦ(G) on high-order perturbations δG in order to minimise dependencies on δG when implementing equation (12) in order to gain a suitable bound κ < 1 [16]. (We additionally make use of closures in order to avoid recomputation of bounds on those subexpressions in the Fréchet derivative DΦ(G)δG that do not depend on δG.)…”

“…We use a method (introduced in [16]), novel in the context of bounding the spectrum of derivatives of renormalisation operators, where we write an eigenvalue, λ, as a linear functional of the corresponding eigenfunction, λ = ϕ(V ), thereby expressing the eigenproblem for (V, λ) as the following nonlinear problem for V ∈ A(Ω):…”