  Let $T$ denote a positive operator with spectral radius $1$ on, say, an
$L^p$-space. A classical result in infinite dimensional Perron--Frobenius
theory says that, if $T$ is irreducible and power bounded, then its peripheral
point spectrum is either empty or a subgroup of the unit circle.
  In this note we show that the analogous assertion for the entire peripheral
spectrum fails. More precisely, for every finite union $U$ of finite subgroups
of the unit circle we construct an irreducible stochastic operator on $\ell^1$
whose peripheral spectrum equals $U$.
  We also give a similar construction for the $C_0$-semigroup case.
