---
abstract: |
    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.
author:
- 'Ian Melbourne [^1]'
- 'Dalia Terhesiu [^2]'
date: '24 August, 2010. Revised 10 August, 2011.'
title: Operator renewal theory and mixing rates for dynamical systems with infinite measure
---

Introduction {#sec-intro}
============

In finite ergodic theory, much recent interest has focussed on the statistical properties of smooth dynamical systems with strong hyperbolicity (expansion/contraction) properties. Landmark results include the proof of exponential decay of correlations for certain classes of uniformly hyperbolic flows 