Jerzy Jezierski

On the growth of the number of periodic points for smooth self-maps of a compact manifold

Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3. We show that in the homotopy class of f there is a C 1 map with less then r periodic points, up to any given fixed period r.

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: NFn(f) = min{#Fix(gn); g ~ f; g continuous} and NJDn(f) = min{#Fix(gn); g ~ f; g smooth}. In general, NJDn(f) may be much greater than NFn(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NFn(f) = NJDn(f) holds for all n ⇔ all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.