When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

There are two algebraic lower bounds of the number of n-periodic points of a self-map f: M → M of a compact smooth manifold of dimension at least 3: NF _n(f) = min{#Fix(g ⁿ); g ~ f; g continuous} and NJD _n(f) = min{#Fix(g ⁿ); g ~ f; g smooth}. In general, NJD _n(f) may be much greater than NF _n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NF _n(f) = NJD _n(f) holds for all n ? all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.