Lusternik-Schnirelmann theory for fixed points of maps

Abstract

We use the ideas of Lusternik-Schnirelmann theory to describe the set
of fixed points of certain homotopy equivalences of a general space.
In fact, we extend
Lusternik-Schnirelmann theory to pairs $(\varphi, f)$, where
$\varphi$ is a homotopy equivalence of a topological space $X$ and where
$f \colon X \rightarrow \mathbb R$ is a continuous function satisfying
$f(\varphi(x)) < f(x)$ unless $\varphi (x) = x$;
in addition, the pair $(\varphi, f)$ is supposed to satisfy a
discrete analogue of the Palais-Smale condition.
In order to estimate the number of fixed points of $\varphi$ in a subset of
$X$, we consider different relative categories.
Moreover, the theory is carried out in an equivariant setting.