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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 C-1 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 C-1 at a boundary of I, then h is C-r in I.