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 \(G \text{-cat} \, f^b -G\text{-cat }\, f^a.\)


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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