We develop a theory of ergodicity for unbounded
functions
 , where
 is a subsemigroup of a locally compact abelian
group 
and  is a Banach space. It is assumed that
is continuous and dominated by a weight
defined on  . In particular, we establish total ergodicity
for the orbits of an (unbounded) strongly
continuous representation
 whose dual representation has no unitary point
spectrum. Under additional conditions stability
of the orbits follows. To study spectra of
functions, we use Beurling algebras
 and obtain new characterizations of their
maximal primary ideals, when
 is non-quasianalytic, and of their minimal
primary ideals, when
 has polynomial growth. It follows that, relative
to certain translation invariant function classes
 , the reduced Beurling spectrum of
 is empty if and only if
 . For the zero class, this is Wiener's tauberian
theorem.
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