Explosion of smoothness for conjugacies between multimodal maps

Let f and g be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy h between f and g is C¹ at a point in the nearby expanding set of f, then h is a smooth diffeomorphism in the basin of attraction of a renormalization interval of f. In particular, if f : I → I and g : J → J are C r unimodal maps and h is C¹ at a boundary of I then h is C^r in I.