The statistics of first returns to decreasing sets
(holes $U_r$)
can be viewed in several ways. One can scale time $t$ as function
of the size $r$ of the hole as suggested by Kac Lemma. This frequently
leads to exponential statistics.
One can also let time go to infinity first, and obtain
the so-called escape rate, and study this rate as the size of the hole
tends to zero.
Both approaches can be seen as special cases of a single scheme in which
$r \to 0$, $t \to \infty$ along varying paths.
In this talk I want to present some results in this direction,
in connection with inducing techniques for interval maps.
This is joint work in progress with Mark Demers (Fairfield) and Mike Todd
(St. Andrews).