We consider a toral Anosov automorphism G γ : T γ → T γ given by G γ (x, y) = (ax+ y, x) in the < v, w > base, where a∈ N\ { 1 } , γ= 1 / (a+ 1 / (a+ 1 / …)) , v= (γ, 1) and w= (- 1 , γ) in the canonical base of R 2 and T γ = R 2 / (vZ× wZ). We introduce the notion of γ-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of G γ ; (ii) affine classes of γ-tilings; and (iii) γ-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.