Periodic points of equivariant maps

Authors

We assume that

$X$ is a compact connected polyhedron,

$G$ is a finite group acting freely on

$X$ , and

$f:X\to X$ a

$G$ -equivariant map. We find formulae for the least number of

$n$ -periodic points in the equivariant homotopy class of

$f$ , i.e.,

$\inf_h |(\mathrm{Fix}(h^n)|$ (where

$h$ is

$G$ -homotopic to

$f$ ). As an application we prove that the set of periodic points of an equivariant map is infinite provided the action on the rational homology of

$X$ is trivial and the Lefschetz number

$L(f^n)$ does not vanish for infinitely many indices

$n$ commeasurable with the order of

$G$ . Moreover, at least linear growth, in

$n$ , of the number of points of period

$n$ is shown.