Slow continued fractions and permutative representations of $\mathcal{O}_N$

TL;DR

This paper constructs representations of the Cuntz algebra $ ext{O}_N$ from interval dynamical systems linked to slow continued fractions, revealing their structure through modular group actions and symmetries.

Contribution

It introduces new representations of $ ext{O}_N$ based on slow continued fraction algorithms and characterizes their irreducible components via modular group symmetries.

Findings

Decomposition formulas for representations are established.

Connections between dynamical systems and automorphisms of $ ext{O}_N$ are demonstrated.

Symmetry properties relate to covariant representations of the flip-flop automorphism.

Abstract

Representations of the Cuntz algebra $\mathcal{O}_N$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^*$--dynamical system ofthe `flip-flop' automorphism of $\mathcal{O}_2$.