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

Grzegorz Graff¹ ; Agnieszka Kaczkowska¹

¹Faculty of Applied Physics and Mathematics Gdańsk University of Technology Narutowicza 11/12 80-233 Gdańsk, Poland

Annales Polonici Mathematici, Tome 107 (2013) no. 1, pp. 29-48

Résumé

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.

Détail

DOI : 10.4064/ap107-1-2

Keywords: smooth self map m dimensional geq closed connected simply connected manifold sequence infty lefschetz numbers its iterations periodic fixed natural wish minimize smooth homotopy class number periodic points periods equal resulting number given topological invariant which defined combinatorial terms constant sufficiently large nbsp compute self maps manifolds simple structure homology groups