  Classical sum rules arise in a wide variety of physical contexts. Asymptotic
expressions have been derived for many of these sum rules in the limit of long
orbital period (or large action). Although sum rule convergence may well be
exponentially rapid for chaotic systems in a global sense with time, individual
contributions to the sums may fluctuate with a width which diverges in time.
Our interest is in the global convergence of sum rules as well as their local
fluctuations. It turns out that a simple version of a lazy baker map gives an
ideal system in which classical sum rules, their corrections, and their
fluctuations can be worked out analytically. This is worked out in detail for
the Hannay-Ozorio sum rule. In this particular case the rate of convergence of
the sum rule is found to be governed by the Pollicott-Ruelle resonances, and
both local and global boundaries for which the sum rule may converge are given.
In addition, the width of the fluctuations is considered and worked out
analytically, and it is shown to have an interesting dependence on the location
of the region over which the sum rule is applied. It is also found that as the
region of application is decreased in size the fluctuations grow. This suggests
a way of controlling the length scale of the fluctuations by considering a time
dependent phase space volume, which for the lazy baker map decreases
exponentially rapidly with time.
