nonuniformly hyperbolic setting.

However, there is a fourth and equally important component, namely

-   An operator renewal equation for flows (Theorem \[thm-renew\]).

As far as we can tell, the operator renewal equation for flows introduced in Section \[sec-renew\] below has no counterpart in the existing probability theory literature. Continuous time versions of renewal theory have been developed previously in the probability theory literature. We refer to [@Feller66 Ch. XI] for the general framework surrounding Blackwell’s renewal theorem [@Blackwell48]. For such a theorem in the infinite mean setting (the continuous time analogue of [@GarsiaLamperti62]) we refer to [@Erickson70 Theorem 1]. We also mention the work of Kingman (see for instance [@Kingman64]) for the continuous analogue, developed for both finite and infinite mean setting, of Feller’s theory on discrete regenerative phenomena. However, it is unclear how to apply these methods here, and our approach seems to have certain advantages as discussed in Remark \[rmk-oprenew\].

In Section \[sec-main\], we state our main results for suspensions of nonuniformly expanding maps, and recover the statements in the introduction (suspensions of intermittent maps) as a special case. The remainder of this paper is then divided into three parts. In Part \[part-oprenew\], we derive the operator renewal 