On graph algebras from interval maps

TL;DR

This paper explores representations of relative graph algebras derived from one-dimensional dynamical systems with escape sets, identifying conditions for their faithfulness based on transitions to escape intervals.

Contribution

It introduces a new class of representations of relative graph algebras linked to dynamical systems with escape sets and characterizes their faithfulness.

Findings

Representations are faithful when transitions to escape intervals satisfy specific conditions.

The study connects dynamical systems with operator algebra representations.

Provides criteria for faithfulness based on orbit transitions.

Abstract

We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of one dimensional dynamical systems, where the underlying Markov interval maps $f$ have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.