Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.10621

On the analytic properties of the perturbing function in the PCR3Body Problem

Corrado Falcolini·Davide Zaccaria
View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide a new expansion of the Fourier coefficient of the Perturbing function of the PCR3Body problem in terms of Hansen Coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e small value of eccentricity and semi-major axis see e.g. \cite{Celletti-Chierchia}). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \Z^2$ and the presence of common zeros between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is a first step for a possible application of \cite{Singular KAM, BBCZ} to PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so called "non--torus" set from $O(1-\sqrt{\e})$ (implied by standard KAM theory) to $O(1-\e |\log\e|^c )$ for some $c>0$.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Corrado Falcolini, Davide Zaccaria. 2025-08-14. On the analytic properties of the perturbing function in the PCR3Body Problem. https://arxiv.org/abs/2508.10621

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↗