Explosion of Smoothness for Conjugacies Between Unimodal Maps

Abstract

Let f and g be C-r unimodal maps, with r >= 3, topologically conjugated by h and without periodic attractors. If h is strongly differentiable at a point p in the expanding set E(f), with h'(p) not equal 0, then, there is an open renormalization interval J such that h is a C-r diffeomorphism in the basin B(J) of J, and h is not strongly differentiable at any point in I \ B(J). The expanding set E(f) contains all points with positive Lyapunov exponent, and if f has a Milnor's interval cycle attractor A then E(f) has full Lebesgue measure.