Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

TL;DR

This paper extends the theory of Cuntz-Krieger algebras for infinite matrices by generalizing branching systems, constructing concrete representations, and analyzing the Perron-Frobenius operator through these representations.

Contribution

It introduces a generalized framework for branching systems and provides explicit matrix representations of the Perron-Frobenius operator for infinite matrices.

Findings

Existence of branching systems for any infinite matrix A

Concrete representations of Cuntz-Krieger algebras using these systems

Matrix representations of the Perron-Frobenius operator under certain conditions

Abstract

In this paper we extend work of Kawamura, see kawamura, for Cuntz-Krieger algebras O_A for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of O_A. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.