Smooth Triangular Maps of the Square with Closed Set of Periodic Points

Let (f, I) and (gx, I) be dynamical systems defined by smooth maps f ∈ C1 (I, I) and gx ∈ C1 (I, I) of the unit interval I = [0, 1]. We consider the triangular map F(x, y) = (f(x), gx(y)) and prove that if every periodic point of f is hyperbolic and the periodic points of F form a closed set, then every nonwandering point of F is periodic.