The central limit theorem for function systems on the torus

The torus satisfies in the central limit theorem for Iterated functions and we can prove the idiea, we do changes on a torus same as the circle but by two open subsets on torus, We can cover a torus with open subsets U and V. Booth U and V are diffeomorphic to a cylinder and therefore have the homotopy type of a circle. We use the homotopy equivalence and equivalence type in the paper of Loring W.TU – An Introduction To Manifolds, on open subsets