## Lusternik-Schnirelmann theory for fixed points of maps.(English)Zbl 1044.37011

The paper contains an interesting generalization of the Lusternik-Schnirelman theory for gradient-like flows. The authors consider homotopy equivalences $$\phi : X\to X$$ from a Hausdorff space into itself, which are “gradient like” i.e., when there exists a function $$f: X \to \mathbb R,$$ bounded below, such that $$f(\phi(x)) \leq f(x)$$ for all $$x \in X.$$
The basic result is as follows: If on the set $$f^b= \{x\in X| f(x)\leq b\}$$:
a) $$f(\phi(x)) < f(x)$$ unless $$\phi(x)=x$$.
b) The pair $$(\phi, f)$$ verifies the following condition of Palais-Smale type: If $$f(x) -f(\phi(x))$$ is not bounded away from zero on a bounded set $$A\subset f^b$$ then $$\phi$$ has a fixed point on the closure of $$A.$$ Then, the category of the set of fixed points of $$\phi$$ on the level set $$f= b$$ is not less than the category of $$f^b.$$
Actually, the paper contains much stronger results. If $$G$$ is a compact Lie group and $$X$$ is a $$G$$ space, then a relative version of the $$G$$-category constructed by the authors in the $$G$$-equivariant setting, under some natural assumptions, allows to estimate from below the number of fixed points of $$\phi$$ on $$f^{-1}([a,b])$$ in terms of $$G \text{-cat} \, f^b -G\text{-cat }\, f^a.$$

### MSC:

 37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems 37B99 Topological dynamics 55M20 Fixed points and coincidences in algebraic topology 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
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