Alcalde Cuesta, Fernando; Bermúdez Carro, Miguel Angel A remark on graphed equivalence relations with harmonic measures. (Une remarque sur les relations d’équivalence graphées, munies de mesures harmoniques.) (French) Zbl 1004.37003 C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 7, 637-640 (2001). Let \((X,{\mathcal B})\) be a standard Borel space, \(\mu\) a measure on \((X,{\mathcal B})\), \({\mathcal R}\) an equivalence relation on \(X\), and let \(\Phi= \{\varphi_i\}\) be a countable set of Borel isomorphisms \(\varphi_i: A_i\to B_i\) between \(A_i\) and \(B_i\) in \({\mathcal B}\). The authors recall several notions in measure and graph theory such as erasings (élagage), graphings (graphage) of \({\mathcal R}\), \(\Phi\)-harmonicity of \(\mu\), Euler characteristic of \(\Phi\) and amenability of \({\mathcal R}\) for \(\mu\). Let \(({\mathcal R},X,\Phi,\mu)\) be a graphed equivalence relation with \(\Phi\)-harmonic measure \(\mu\). The authors give, among other things, two criteria for \({\mathcal R}\) to be amenable for \(\mu\). Reviewer: A.Morimoto (Nagoya) Cited in 1 Document MSC: 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations Keywords:harmonicity; amenability; graphed equivalence relation PDFBibTeX XMLCite \textit{F. Alcalde Cuesta} and \textit{M. A. Bermúdez Carro}, C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 7, 637--640 (2001; Zbl 1004.37003) Full Text: DOI