  We develop a theory of operator renewal sequences in the context of infinite
ergodic theory. For large classes of dynamical systems preserving an infinite
measure, we determine the asymptotic behaviour of iterates $L^n$ of the
transfer operator. This was previously an intractable problem.
  Examples of systems covered by our results include (i) parabolic rational
maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly
expanding interval maps with indifferent fixed points.
  In addition, we give a particularly simple proof of pointwise dual ergodicity
(asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under
consideration.
  In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain
error estimates in the associated Dynkin-Lamperti arcsine laws.
