### Lusternik-Schnirelmann theory for fixed points of maps

DOI: http://dx.doi.org/10.12775/TMNA.2003.011

#### 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.

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.

#### Keywords

Fixed point theory; Lusternik-Schnirelman theory; Palais-Smale condition

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