| Popis: |
We analyse Haar systems associated to groupoids obtained by certain equivalence relations of dynamical nature on sets like \(\{1,2,...,d\}^\mathbb {Z}\), \(\{1,2,...,d\}^\mathbb {N}\), \(S^1\times S^1\), or \((S^1)^\mathbb {N}\), where \(S^1\) is the unitary circle. We also describe properties of transverse functions, quasi-invariant probabilities and KMS states for some examples of von Neumann algebras (and also \(C^*\)-Algebras) associated to these groupoids. We relate some of these KMS states with Gibbs states of Thermodynamic Formalism. We will show new results but we will also describe in detail several examples and basic results on the above topics. Some known results on non-commutative integration are presented, more precisely, the relation of transverse measures, cocycles and quasi-invariant probabilities. We describe the results in a language which is more familiar to the people in Dynamical Systems. Our intention is to study Haar systems, quasi-invariant probabilities and von Neumann algebras as a topic on measure theory (intersected with ergodic theory) avoiding questions of algebraic nature (which, of course, are also extremely important) |
