MINIMAL NUMBER OF PERIODIC POINTS FOR SMOOTH SELF-MAPS OF S 3 GRZEGORZ GRAFF AND JERZY JEZIERSKI Abstract. Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3, r a fixed natural number. A topological invariant D m r [f ], introduced in [5], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f . In this paper we calculate D 3 r [f ] for all self-maps of S 3 . 1. Introduction The classical problem in periodic point theory is to determine or esti- mate the least number of r-periodic points in homotopy class of a given f , a self-map of a compact manifold M m , where r is a fixed natural number (cf. [10]). If M m is simply-connected and has the dimension m ≥ 3, then one can always find a map g homotopic to f with only one point in Fix(g r ) (cf. [9]). This is however impossible, if we demand additionally that g is smooth. In [5] the authors define the topological invariant D m r [f ] equal to the minimal number of elements in Fix(g r ) for all g which are smooth and homotopic to f . This enables to explore the new smooth branch of Nielsen periodic point theory. Let us remark that D m r [f ] may be inter- preted purely in the terms of smooth category, namely we may assume that f is smooth and approximate the homotopy which joins f with g by a smooth one. Then D m r [f ] gives the minimal number of periodic points in smooth homotopy class of the given f . This invariant is obtained by decomposing Lefschetz numbers of iter- ations into sequences which can be locally realized as fixed point indices of iterations at an isolated periodic orbit for some C 1 map. As a result, to find the exact value of D m r [f ] we need two types of data: the informa- tion about the form of {L(f n )} n|r (more precisely of the set of so-called 2000 Mathematics Subject Classification. Primary 37C25, 55M20; Secondary 37C05. Key words and phrases. Least number of periodic points, indices of iterations, smooth maps, low dimensional dynamics. Research supported by KBN grant No 1 P03A 03929. 1