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

Abstract. 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 λ 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 λ as eigenvalue of all the maps f∗k with k odd is equal to the sum of the multiplicities of λ 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.