Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2409.20457

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

Maxime Breden·Jorge Gonzalez·J. D Mireles James
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maxime Breden, Jorge Gonzalez, J. D Mireles James. 2024-09-30. Validated enclosure of renormalization fixed points via Chebyshev series and the DFT. https://arxiv.org/abs/2409.20457

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗