[@Chernov98; @Dolgopyat98; @Liverani04] and planar dispersing billiards [@Young98]. The latter is part of a general scheme [@Young98; @Young99] for estimating decay of correlations, or mixing rates, for discrete time dynamical systems.

For systems with subexponential decay of correlations, most approaches yielded only upper bounds for mixing rates. Sarig [@Sarig02] introduced a powerful new technique, [*operator renewal theory*]{}, to obtain precise asymptotics and hence sharp mixing rates. This is an extension of scalar renewal theory from probability theory to the operator situation. The technique was substantially extended and refined by Gouëzel [@Gouezel04; @Gouezel05].

Garsia & Lamperti [@GarsiaLamperti62] developed a theory of renewal sequences with infinite mean in the probabilistic setting. The techniques are very different from the finite mean case, and draw heavily on the theory of regular variation [@BinghamGoldieTeugels; @Feller66; @Karamata33]. A natural question is to develop renewal operator theory in the infinite mean case.

#### Renewal sequences

In probability theory, renewal sequences relate return probabilities to a specified “nice” set with first return probabilities. The analogue in ergodic theory arises in the study of first return maps.

Let $(X,\mu)$ be a measure space (finite or infinite), and $f:X\to X$ a conservative measure preserving map. Fix $Y\subset X$ with $\mu(Y)>0$. 