C¹ self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

© 2016, Akademie ved Ceske Republiky. All rights reserved. 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 C1 having finitely many periodic points all of them hyperbolic, then f is almost quasi-unipotent.