The Hopf decomposition of locally compact group actions

TL;DR

This work develops a comprehensive, unified theory of the Hopf decomposition for nonsingular actions of locally compact second countable groups, extending classical results from countable groups and single transformations to the lcsc setting. It systematizes four interrelated viewpoints—Hopf’s formulation via return functions, recurrence properties, smooth orbit equivalence relations, and Maharam extensions—and proves a structure theorem for totally dissipative actions, including a Krengel-type description in terms of random compact subgroups. The paper also shows that dissipativity interacts coherently with ergodic decomposition and subactions of closed subgroups, yielding translation and coset-space models (Krengel spaces) that capture the dissipative locus. Further results connect the conservativity of Maharam extensions to cocycle recurrence and establish maximal transient sets and positive–null decompositions, offering a robust framework for analyzing group actions beyond the free/countable case. These contributions provide foundational tools for ergodic theory, homogeneous dynamics, and operator-algebraic contexts where lcsc groups act nonsingularly.

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.

Figures (2)

• Figure 1: An arrow presents the passage from class $A$ of totally dissipative $G$-spaces to its subclass $B$ that satisfies the restrictive property P.
• Figure 2: The relations between the Hopf decomposition $X=\mathcal{C}\sqcup\mathcal{D}$ and the positive--null decomposition $X=\mathcal{P}_{1}\sqcup\mathcal{P}_{\infty}\sqcup\mathcal{N}$. Unless $G$ is compact, by the Poincaré recurrence theorem \ref{['thm:poincareacip']} and by Proposition \ref{['prop:dissacim']}, we have $\mathcal{D}\subseteq\mathcal{P}_\infty$.

Theorems & Definitions (81)

• Lemma 2.1: The exhaustion lemma
• Proposition 2.2
• Theorem A: Hopf decomposition
• Theorem B: The recurrence theorem
• Theorem B'
• Theorem 4.1
• Definition 4.2
• Proposition 4.3
• proof : Proof of Proposition \ref{['prop:recurrence']}
• proof : Proof of Theorem \ref{['thm:recurrence']}