Title: Haar systems, KMS states on von Neumann algebras and $C^*$-algebras on dynamically defined groupoids and Noncommutative Integration,
Abstract: We analyse certain Haar systems associated to groupoids obtained by certain
natural 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. While presenting new results, we will also
describe in detail several examples and basic results on the above topics. In
other words it is also a survey paper. 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 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).