Thermodynamic formalism for Haar systems in noncommutative integration: transverse functions and entropy of transverse measures

Abstract

We generalize to Haar Systems on groupoids some results related to entropy and pressure which are well known in Thermodynamic Formalism. Given a Haar system, we introduce a transfer operator, similar to the Ruelle operator, where the equivalence relation (the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the a priori probability. We introduce the concept of invariant transverse probability and also of entropy for an invariant transverse probability as well as the pressure for transverse functions. We also explore the relation between quasi-invariant probabilities and transverse measures. In a section of examples the Rokhlin's disintegration theorem will play an important role when considering a large class of dynamically defined groupoids.

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

“…In addition to being connected with the previous work [27], we remark that there are at least two natural reasons for our preference of the above approach using probability kernels instead of a probability π 0 . The first one is because the conditional relative entropy, as above defined, does not consider π 0 totally, but only ν, while the y−marginal Q of π 0 is replaced by Q.…”

“…where ν is an a priori probability on M , µ is a shift-invariant (stationary) probability on Ω := M N = {|x 1 , x 2 , x 3 , ...)|x i ∈ M ∀i ∈ N} and the functions c : M N → R are necessarily Lipschitz. Variations of this expression appear in [29], [36] and more recently in [27].…”

“…Following [27] we consider for the present setting the definition of entropy described below. Definition 1.2.…”

“…Following [27], it is possible to remark that there are natural extensions of the above concepts if we replace X × Y by a measurable space M with a measurable partition (which induces an equivalence relation) and probability kernels by general transverse functions. On the other hand, the above information gain is related with the generalized conditional relative entropy (see chap.…”

“…Our purpose in this section is to extend the definition of information gain IG(π, π 0 ), given by ( 19), for the case when X and Y are measurable spaces. As we will see, a natural way of to extend (19) is by considering probability kernels and the notion of entropy given in [27]. This entropy is an extension of that previously introduced in [26] and [36] for compact spaces using an a priori probability.…”