Minimal number of periodic points for smooth self-maps of compact manifolds
by
Grzegorz Graff
Gdansk University of Technology, Faculty of Mathematics and Physics
Coauthors: Jerzy Jezierski, Institute of Applications of Mathematics, Agricultural University of Warsaw (SGGW)

Let f be a self-map of a compact simply-connected manifold of dimension m ≥ 3, r be a fixed natural number. In this talk two results, obtained by a use of Fixed Point Index and Nielsen periodic point theory, will be presented.

(1) We define the topological invariant D^m_r[f] which is the best lower bound for the number of r-periodic points for all smooth maps homotopic to f. In case m=3 we give the formula for D³_r[f] and calculate it for some types of manifolds.

(2) In 1974 Shub and Sullivan conjectured that the growth of periodic points of a smooth map f with unbounded Lefschetz numbers is (asymptotically) exponential. We show that in homotopy class of f there is a smooth map g such that # (∪_{i ≤ r}Fix(gⁱ)) ≤ r-1.

Date received: February 24, 2006