A sufficient condition for the realizability of the least number of periodic points of a smooth map

Abstract

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). In the simply connected case, the equality of the two numbers is equivalent to the sequence of Lefschetz numbers satisfying restrictions introduced by Chow, Mallet-Parret and Yorke (1983). The last condition is not sufficient in the non-simply connected case. Here we give some conditions which guarantee the equality when π1M= Z2.