Validated enclosure of renormalization fixed points via Chebyshev series and the DFT

TL;DR

This paper introduces a computational framework using Chebyshev series and interval arithmetic in Julia to rigorously prove the existence, uniqueness, and stability of renormalization fixed points for Feigenbaum-Cvitanovi7ic operators, achieving high-precision results.

Contribution

It develops a novel Chebyshev series-based discretization method for renormalization operators, enabling rigorous proofs and high-precision computations of fixed points and universal constants.

Findings

Proved existence of multiple fixed points for orders 3 to 10.

Validated bounds on universal constants for these fixed points.

Reproved the classical Feigenbaum fixed point with nearly 500 decimal digits accuracy.

Abstract

This work develops a computational framework for proving existence, uniqueness, isolation, and stability results for real analytic fixed points of $m$-th order Feigenbaum-Cvitanovi\'{c} renormalization operators. Our approach builds on the earlier work of Lanford, Eckman, Wittwer, Koch, Burbanks, Osbaldestin, and Thurlby \cite{iii1982computer,eckmann1987complete,MR0727816, burbanks2021rigorous2,burbanks2021rigorous1}, however the main point of departure between ours and previous studies is that we discretize the domain of the renormalization operators using Chebyshev rather than Taylor series. The advantage of Chebyshev series is that they are naturally adapted to spaces of real analytic functions, in the sense that they converge on ellipses containing real intervals rather than on disks in $\mathbb{C}$. The main disadvantage of working with Chebyshev series in this context is that the essential operations of rescaling and composition are less straight forward for Chebysehv than for Taylor series. These difficulties are overcome using a combination of a-priori information about decay rates in the Banach space with a-posteriori estimates on Chebyshev interpolation errors for analytic functions. Our arguments are implemented in the Julia programming language and exploit extended precision floating point interval arithmetic. In addition to proving the existence of multiple renormalization fixed points of order $m = 3, \ldots, 10$, and computing validated bounds on the values of their the universal constants, we also reprove the existence of the classical $m=2$ Feigenbaum renormalization fixed point and compute its universal constants to close to 500 correct decimal digits.