equation for flows. In Part \[part-infinite\], we prove our results on infinite measure systems. In Part \[part-finite\], we prove our results on finite measure systems. The paper is written in such a way that Parts \[part-infinite\] and \[part-finite\] can be read independently.

For discrete time systems, operator renewal theory was developed first in the finite measure case before being extended to the infinite measure situation. For continuous time systems, we present the material in the reverse order. The reason for this is that having formulated the continuous time operator renewal equation described above in (iv), it is fairly straightforward to deduce our main results for infinite measure semiflows from the existing work described in components (ii) and (iii). (We note however that certain technical estimates in the proof and usage of Lemma \[lem-R\] are considerably more complicated than in the case of discrete time, infinite measure.) In contrast, although our results for finite measure semiflows follow from components (i), (iii) and (iv), it requires significantly more work to glue these methods together.

#### Notation

We use the “big $O$” and $\ll$ notation interchangeably, writing $a_n=O(b_n)$ or $A_n\ll b_n$ if there is a constant $C>0$ such that $a_n\le Cb_n$ for all 