Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.09093

On stationary actions of locally compact groups and their Radon-Nikodym cocycles

Nachi Avraham-Re'em·Michael Björklund
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study stationary actions of locally compact measured groups through the structure and regularity of their Radon-Nikodym cocycles. We start with two dynamical consequences of stationarity. Extending a theorem of Furstenberg-Glasner from discrete groups to noncompact locally compact groups, we show that every stationary action is conservative. Thus stationary actions are never of type I. We then show that an ergodic stationary action admitting an absolutely continuous invariant sigma-finite measure is in fact probability preserving. Thus stationary actions are never of type II_infty. Using a construction of Katznelson-Weiss and Vaes-Verjans, we show that if a group admits a stationary action of type III_1, then it admits stationary actions of every type III_lambda. The second part concerns the regularity of the Radon-Nikodym cocycle. We introduce the harmonic majorant on normalized positive harmonic functions, which gives Harnack-type control of Radon-Nikodym derivatives of stationary actions. For compactly supported probability measures with an Lp-density for some p>1, we prove that the harmonic majorant is finite and locally bounded. As a consequence, such measured groups admit a universal compact Radon-Nikodym model: a single compact G-space with a continuous cocycle into which stationary actions can be realized, so that the ambient cocycle gives a version of its Radon-Nikodym cocycle. This strengthens the Mackey-Varadarajan compact model theorem by incorporating the Radon-Nikodym cocycle into the model. By contrast, we construct a random walk on the real affine group whose Poisson boundary fails Kaimanovich's SAT* property: the Poisson kernel is unbounded arbitrarily close to the identity. Therefore, Harnack's inequality already fails for positive harmonic functions. In particular, this Poisson boundary admits no topological model with continuous Poisson kernel.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nachi Avraham-Re'em, Michael Björklund. 2026-04-10. On stationary actions of locally compact groups and their Radon-Nikodym cocycles. https://arxiv.org/abs/2604.09093

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↗