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

TL;DR

This paper establishes precise, computer-assisted bounds on the renormalisation fixed point and eigenfunctions for period doubling in unimodal maps with degree 4 critical points, advancing the mathematical understanding of these systems.

Contribution

It introduces a rigorous, computer-assisted method to tightly bound the renormalisation fixed point and eigenfunctions for maps with degree 4 critical points, using contraction mapping and multi-precision arithmetic.

Findings

Tight bounds on the renormalisation fixed point for degree 4 maps.

Precise eigenvalues and eigenfunctions with over 320 significant figures.

Validated the universality constants with high-precision computations.

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 use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation 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.