# Thermodynamic Formalism for Haar systems in Noncommutative Integration: transverse functions and entropy of transverse measures

## TL;DR

This paper extends thermodynamic formalism to Haar systems in groupoids derived from equivalence relations, introducing concepts like transfer operators, invariant transverse probabilities, and entropy within a noncommutative integration framework.

## Contribution

It generalizes entropy and pressure concepts to Haar systems in groupoids, introducing transfer operators and invariant transverse probabilities in a noncommutative setting.

## Key findings

- Defined a transfer operator based on the equivalence relation
- Introduced invariant transverse probability and entropy concepts
- Explored relations between quasi-invariant probabilities and transverse measures

## Abstract

We consider here a class of groupoids obtained via an equivalence relation (the subgroupoids of pair groupoids). We generalize to Haar Systems in these groupoids some results related to entropy and pressure which are well known in Thermodynamic Formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the {\it a priori} probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Our results are on measurable category.