Fixed Point Theory and Applications 
Volume 2006 (2006), Article ID 29470, 10 pages
doi:10.1155/FPTA/2006/29470

Epsilon Nielsen fixed point theory

Robert F. Brown

 Department of Mathematics, University of California, Los Angeles 90095-1555, CA, USA
Received 11 October 2004; Revised 17 May 2005; Accepted 21 July 2005

Abstract

Let f:X→X be a map of a compact, connected Riemannian manifold, with or without boundary. For ∈>0 sufficiently small, we introduce an ∈-Nielsen number N∈(f) that is a lower bound for the number of fixed points of all self-maps of X that are ∈-homotopic to f. We prove that there is always a map g:X→X that is ∈-homotopic to f such that g has exactly N∈(f) fixed points. We describe procedures for calculating N∈(f) for maps of 1-manifolds.