Rigidity in one-dimensional tiling spaces

Suppose T φ and T θ are tiling spaces arising from primitive nonperiodic substitutions φ and θ. Suppose F φ and F θ denote the corresponding inflation and substitution maps on the respective tiling spaces. We prove that T φ and T θ are homeomorphic if and only if there exist positive integers m and n such that F φ n and F θ m are topologically conjugate.