EIGENVALUES AND DYNAMICAL DEGREES OF SELF-MAPS ON ABELIAN VARIETIES

Let X be a smooth projective variety over an algebraically closed field, and f : X -> X a surjective self-morphism of X. The i-th cohomological dynamical degree chi i(f) is defined as the spectral radius of the pullback f* on the e ' tale cohomology group Hie ' t(X,Qe) and the k-th numerical dynamical degree lambda k(f) as the spectral radius of the pullback f* on the vector space Nk(X)R of real algebraic cycles of codimension k on X modulo numerical equivalence. Truong conjectured that chi 2k(f) = lambda k(f) for all 0 < k < dim X as a generalization of Weil's Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self maps of abelian varieties in prime characteristic, which is of independent interest.