Ergodic Theory and Dynamical Systems

C*-algebras, approximately proper equivalence relations and thermodynamic formalism

R. EXEL a1 and A. LOPES a2
a1 Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, Brazil (e-mail:
a2 Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil (e-mail:

Article author query
exel r   [Google Scholar] 
lopes a   [Google Scholar] 


We introduce a non-commutative generalization of the notion of (approximately proper) equivalence relations and propose the construction of a ‘quotient space’. We then consider certain one-parameter groups of automorphisms of the resulting C*-algebra and prove the existence of KMS states at every temperature. In a model originating from thermodynamics we prove that these states are unique as well. We also show a relationship between maximizing measures (the analogue of the Aubry–Mather measures for expanding maps) and ground states. In the last section we explore an interesting example of phase transitions.

(Received July 29 2002)
(Revised April 29 2003)