$C^{\ast }$-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

We study the $C^{\ast }$-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 $C^{\ast }$-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.