  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.
