Epsilon Nielsen fixed point theory (Q2491513)

Let \(X\) be a compact connected Riemannian manifold (possibly with boundary) and \(f:X\to X\) a map. Let \(\varepsilon>0\) be so small that points within a distance of \(\varepsilon\) are connected by a unique geodesic. Denote by \(MF^\varepsilon(f)\) the minimum number of fixed points for maps \(g:X\to X\) which are \(\varepsilon\)-homotopic to \(f\). Fixed points \(x,y\) of \(f\) are \(\varepsilon\)-equivalent if they are contained in the same component of \(\Delta^\varepsilon(f):=\{z\in X| \;d(z,f(z))<\varepsilon\}\). There are finitely many \(\varepsilon\)-equivalence classes. A class \(\mathbb{F}\) is called essential if the fixed point index \(i(f,\Delta)\not=0\) where \(\Delta\) is the component of \(\Delta^\varepsilon(f)\) containing \(\mathbb{F}\). The \(\varepsilon\)-Nielsen number of \(f\), \(N^\varepsilon(f)\), is defined to be the number of essential \(\varepsilon\)-classes. The author proves that \(\varepsilon\)-equivalent fixed points belong to the same Nielsen fixed point class and one has that \(N(f)\leq N^\varepsilon(f)\leq MF^\varepsilon(f)\). The author's main result states that for every \(f:X\to X\) there exists a \(g:X\to X\) which is \(\varepsilon\)-near to \(f\) such that \(g\) has exactly \(N^\varepsilon(f)\) fixed points. Finally the author describes a procedure that determines \(N^\varepsilon(f)\) in case \(\dim X=1\).