Continuous groupoids on the symbolic space, quasi-invariant probabilities for Haar systems and the Haar-Ruelle operator

TL;DR

This paper generalizes the Ruelle operator to a Haar-Ruelle operator for groupoids on symbolic spaces, establishing connections between quasi-invariant probabilities and eigenprobabilities, with applications to H"older cocycles and IFS with weights.

Contribution

It introduces the Haar-Ruelle operator incorporating Haar structure and extends the theory to continuous and H"older cocycles, broadening the scope of thermodynamic formalism.

Findings

Generalization of the Ruelle operator to Haar-Ruelle operator.

Establishment of links between quasi-invariant and eigenprobabilities.

Application to H"older cocycles and iterated function systems with weights.

Abstract

We consider groupoids on $\{1,2,..,d\}^\mathbb{N}$, cocycles and the counting measure as transverse function. We generalize results relating quasi-invariant probabilities with eigenprobabilities for the dual of the Ruelle operator. We assume a mild compatibility of the groupoid with the symbolic structure. We present a generalization of the Ruelle operator - the Haar-Ruelle operator - taking into account the Haar structure. We consider continuous and also H\"older cocycles. IFS with weights appears in our reasoning in the H\"older case.