C-1 SELF-MAPS ON CLOSED MANIFOLDS WITH FINITELY MANY PERIODIC POINTS ALL OF THEM HYPERBOLIC

Let X be a connected closed manifold and f a self- map on X. We say that f is almost quasi- unipotent if every eigenvalue lambda of the map f(*k) (the induced map on the k-th homology group of X) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of lambda as eigenvalue of all the maps f(*k) with k odd is equal to the sum of the multiplicities of lambda as eigenvalue of all the maps f(*k) with k even. We prove that if f is C-1 having finitely many periodic points all of them hyperbolic, then f is almost quasi-unipotent.