The Hopf decomposition of locally compact group actions

TL;DR

This paper extends the classical Hopf Decomposition to actions of locally compact second countable groups, introducing new techniques for understanding recurrence, transience, and stabilizer structures.

Contribution

It provides a unified framework and new characterizations for the Hopf Decomposition in the context of general group actions, generalizing previous results for flows.

Findings

New structural insights into recurrence and transience.

Characterizations of dissipative actions.

Generalization of Krengel's classical result.

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.