Chaotic attractors

Let f and g be piecewise smooth interval maps, with critical-singular sets, and A a cycle of intervals for f . We prove that A is a topological chaotic attractor if, and only if, A is a metric chaotic attractor. Let h|A be a topological conjugacy between f and g. We prove that, if h is differentiable at a single point p of the visiting set V , with non zero derivative, then h is smooth in A. Furthermore, the visiting set V is a residual set of A and, if the sets Cf and Cg are critical then V has µ full measure, for every expanding measure µ, with supp µ = A