Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1812.07511

Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms

Gábor Elek
View original source ↗Read source document ↗Export metadata ↓

Abstract

The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and local-global graph limit theories for sparse graphs. The natural limit objects are continuous actions of finitely generated groups on totally disconnected compact metric spaces. We prove that the space of weak equivalent classes of free Cantor actions is compact and contains a smallest element, as in the measurable case. We will introduce and study various notions of almost finiteness, the qualitative analogue of hyperfiniteness, for classes of bounded degree graphs. We prove the almost finiteness of a new class of \'etale groupoids associated to Cantor actions and construct an example of a nonamenable, almost finite totally disconnected \'etale groupoid, answering a query of Suzuki. Motivated by the notions and results on qualitative graph limits, in the second part of our paper we give a precise definition of constant-time distributed algorithms on sparse graphs. We construct such constant-time algorithms for various approximation problems for hyperfinite and almost finite graph classes. We also prove the Hausdorff convergence of the spectra of convergent graph sequences in the strongly almost finite category.

Explore related subjects

math.DScs.DCmath.CO

Keep this discovery

BibTeXRIS

Gábor Elek. 2018-12-18. Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms. https://arxiv.org/abs/1812.07511

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↗