Cocycles on groupoids arising from -actions
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...
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| Published in | Ergodic theory and dynamical systems Vol. 42; no. 11; pp. 3325 - 3356 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
01.11.2022
|
| Online Access | Get full text |
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| Summary: | 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 and uniqueness results for KMS states associated to the graphs. |
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| ISSN: | 0143-3857 1469-4417 |
| DOI: | 10.1017/etds.2021.69 |