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We develop a theory of operator renewal sequences in the context of infiniteergodic theory. For large classes of dynamical systems preserving an infinitemeasure, we determine the asymptotic behaviour of iterates $L^n$ of thetransfer operator. This was previously an intractable problem.  Examples of systems covered by our results include (i) parabolic rationalmaps of the complex plane and (ii) (not necessarily Markovian) nonuniformlyexpanding 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 underconsideration.  In certain situations, including Pomeau-Manneville intermittency maps, weobtain higher order expansions for $L^n$ and rates of mixing. Also, we obtainerror estimates in the associated Dynkin-Lamperti arcsine laws.

Operator renewal theory and mixing rates for dynamical systems with infinite measure