The Hopf decomposition of locally compact group actions
Dynamical Systems
2026-01-28 v3
Abstract
We develop a unified approach to the classical Hopf Decomposition (also known as the conservative--dissipative decomposition) for actions of locally compact second countable groups. While the decomposition is well understood for free actions of countable groups, the extension to general actions requires new techniques and structural insights, particularly concerning recurrence and transience, cocycle behavior, and the structure of stabilizers. We establish several new characterizations and prove a structure theorem for totally dissipative actions, generalizing Krengel's classical result for flows.
Cite
@article{arxiv.2406.17137,
title = {The Hopf decomposition of locally compact group actions},
author = {Nachi Avraham-Re'em and George Peterzil},
journal= {arXiv preprint arXiv:2406.17137},
year = {2026}
}
Comments
35 pages, 2 figures. Change of title and additional minor changes. A version accepted in Journal d'Analyse Mathematique