C*-algebras of commuting endomorphisms

TL;DR

This paper constructs C*-algebras from commuting endomorphisms on compact spaces, capturing the dynamics of N^2-actions, and extends to various examples including rank two graphs and subshifts.

Contribution

It introduces a new C*-algebra framework for commuting endomorphisms, using Cuntz-Pimsner algebras for general cases beyond local homeomorphisms.

Findings

Constructed C*-algebras reflect N^2-dynamics.

Extended the framework to rank two graphs and subshifts.

Connected algebraic structures to dynamical systems.

Abstract

Given a compact space X and two commuting continuous open surjective maps sigma_1, sigma_2 : X --> X, we construct certain C*-algebras that reflect the dynamics of the N^2-action. When the maps sigma_1, sigma_2 are local homeomorphisms, these are groupoid algebras, but in general, we will use a Cuntz-Pimsner algebra associated to a product system of Hilbert bimodules in the sense of Fowler. The motivating example for our construction is the dynamical system associated with a rank two graph by Kumjian and Pask. We consider also a two-dimensional subshift of Ledrappier, the case of two covering maps of the circle, and the two-dimensional Bernoulli shift.