Least number of periodic points of self-maps of Lie groups

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 is continuous} and NJD n (f) = min{#Fix(g n ); g ∼ f; g is smooth}. In general, NJD n (f) may be much greater than NF n (f). If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds ⇔ all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.