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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 .

Explosion of smoothness for conjugacies between multimodal maps