On Mañé.s Proof of the C 1 Stability Conjecture

It seems that in Mañé.s proof of the C 1.-stability conjecture containing in the famous paper which published in I. H. E. S. (1988), there exists a deficiency in the main lemma which says that for f. F 1(M) there exists a dominated splitting TM..P....i(f)=E.si.F.ui (0 < i < dim M) such that if E.si is contracting, then F.ui is expanding. In the first part of the paper, we give a proof to fill up this deficiency. In the last part of the paper, we, under a weak assumption, prove a result that seems to be useful in the study of dynamics in some other stability context.