  The spectral theory of semigroup generators is a crucial tool for analysing
the asymptotic properties of operator semigroups. Typically, Tauberian
theorems, such as the ABLV theorem, demand extensive information about the
spectrum to derive convergence results. However, the scenario is significantly
simplified for positive semigroups on Banach lattices. This observation extends
to the broader class of eventually positive semigroups -- a phenomenon observed
in various concrete differential equations. In this paper, we investigate the
spectral and asymptotic properties of eventually positive semigroups, focusing
particularly on the persistently irreducible case. Our findings expand upon the
existing theory of eventual positivity, offering new insights into the
cyclicity of the peripheral spectrum and asymptotic trends. Notably, several
arguments for positive operators and semigroups do not apply in our context,
necessitating the use of ultrapower arguments to circumvent these challenges.
