See M. Herman, Construction d’un diffeomorphisme minimale d’entropie nulle,
Erg. Theory and Dyn. Syst. 1 (1981) 65-76
bonatti (2017-06-17 15:44:40 -0700 -0700):
In dimension >2 the answer is “no”: the product of any diffeomorphism f by an
irrational rotation (or translation) as no periodic points, hence is Kupka-
Smale, but has the same entropy as f (hence it can be >0).
Comments
See M. Herman, Construction d’un diffeomorphisme minimale d’entropie nulle, Erg. Theory and Dyn. Syst. 1 (1981) 65-76
In dimension >2 the answer is “no”: the product of any diffeomorphism f by an irrational rotation (or translation) as no periodic points, hence is Kupka- Smale, but has the same entropy as f (hence it can be >0).