Continuous Groupoids on the Symbolic Space, Quasi-Invariant Probabilities for Haar Systems and the Haar–Ruelle Operator

  • Artur O. LopesEmail author
  • Elismar R. Oliveira


We consider groupoids on \(\{1,2,\ldots ,d\}^\mathbb {N}\), cocycles and the counting measure as transverse function. We generalize results relating quasi-invariant probabilities with eigenprobabilities for the dual of the Ruelle operator. We assume a mild compatibility of the groupoid with the symbolic structure. We present a generalization of the Ruelle operator—the Haar–Ruelle operator—taking into account the Haar structure. We consider continuous and also Hölder cocycles. IFS with weights appears in our reasoning in the Hölder case.


Groupoid Continuos groupoids Quase-invariant probabilities Haar systems Ruelle-operator Haar-Ruelle operator IFS KMS state von Neumann Algebra Cocycle Transverse measure 


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© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Instituto de Matematica e Estatistica-UFRGSPorto AlegreBrazil

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