$C^*$-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

TL;DR

This paper investigates the $C^*$-algebras arising from asymptotic equivalence relations of hyperbolic toral automorphisms, revealing they are isomorphic to four-dimensional non-commutative tori and analyzing their $K_0$-groups.

Contribution

It explicitly computes the $C^*$-algebras associated with hyperbolic automorphisms on the torus and characterizes their $K_0$-group traces in terms of the automorphisms.

Findings

The $C^*$-algebras are four-dimensional non-commutative tori.

The traces on $K_0$-groups are described via hyperbolic matrices.

Explicit numerical computations support the theoretical results.

Abstract

We study the $C^*$-algebras of the \'etale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $K_0$-groups of the $C^*$-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.