Cocycles on Certain Groupoids Associated to $ \mathbb{N}^{k} $-Actions

TL;DR

This paper studies groupoids from commuting local homeomorphisms on compact spaces, characterizes 1-cocycles, extends Ruelle-Perron-Frobenius theory, and analyzes KMS states, especially in higher-rank graph contexts.

Contribution

It introduces a new characterization of 1-cocycles on these groupoids and extends Ruelle-Perron-Frobenius theory to systems with multiple commuting operators.

Findings

New characterization of 1-cocycles via continuous functions.

Extended Ruelle-Perron-Frobenius theory for k-operator systems.

Results on existence and uniqueness of KMS states for associated C*-algebras.

Abstract

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{\ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast} $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.