Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers
Volume 107 / 2013
Annales Polonici Mathematici 107 (2013), 29-48
MSC: Primary 37C25, 55M20; Secondary 37C05.
DOI: 10.4064/ap107-1-2
Abstract
Let $f$ be a smooth self-map of an $m$-dimensional ($m\geq 4$) closed connected and simply-connected manifold such that the sequence $\{L(f^n)\}_{n=1}^{\infty}$ of the Lefschetz numbers of its iterations is periodic. For a fixed natural $r$ we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to $r$. The resulting number is given by a topological invariant $J[f]$ which is defined in combinatorial terms and is constant for all sufficiently large $r$. We compute $J[f]$ for self-maps of some manifolds with simple structure of homology groups.
