Cocycles on groupoids arising from -actions

Carla Farsi,Judith Packer,Leonard Huang,Alex Kumjian

doi:10.1017/etds.2021.69

Abstract

AbstractWe 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 and uniqueness results for KMS states associated to the graphs.

Highlights

Let X be a compact Hausdorff space and σ : X → X a local homeomorphism of X onto itself
The étale groupoid associated to a finite family of commuting local homeomorphisms of a compact metric space has gone by various names including Deaconu–Renault groupoids of higher rank and the semidirect product groupoid corresponding to the action of the semigroup Nk [10]
We study KMS states for groupoids associated to a finite family of commuting local homeomorphisms of a compact metric space by further developing a Ruelle–Perron–Frobenius (RPF) theory of dynamical systems of several commuting operators

Summary

Introduction

Let X be a compact Hausdorff space and σ : X → X a local homeomorphism of X onto itself. Our objective is to give an algebraic way of constructing all continuous H -valued 1-cocycles, where H is a locally compact abelian group, on groupoids associated to a finite family of commuting local homeomorphisms of a compact metric space.

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Results

Conclusion