  This paper considers strongly continuous semigroups of operators on Banach
lattices which are locally eventually positive, a property that was first
investigated in the context of concrete fourth-order evolution equations. We
construct a simple example to show that the typical assumptions on the spectrum
of the semigroup generator considered currently in the literature are far from
necessary in the more general setting of local eventual positivity. Under
minimal additional assumptions, we obtain results on the asymptotic behaviour
of orbits, as well as necessary conditions on the peripheral point spectrum of
locally eventually positive semigroups.
