Papers › Validated enclosure of renormalization fixed points via Chebyshev series and the DFT
Validated enclosure of renormalization fixed points via Chebyshev series and the DFT
Maxime Breden, Jorge Gonzalez, J. D Mireles James
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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 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, …, 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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections