  We study Abelian sandpiles on graphs of the form $G \times I$, where $G$ is
an arbitrary finite connected graph, and $I \subset \Z$ is a finite interval.
We show that for any fixed $G$ with at least two vertices, the stationary
measures $\mu_I = \mu_{G \times I}$ have two extremal weak limit points as $I
\uparrow \Z$. The extremal limits are the only ergodic measures of maximum
entropy on the set of infinite recurrent configurations. We show that under any
of the limiting measures, one can add finitely many grains in such a way that
almost surely all sites topple infinitely often. We also show that the extremal
limiting measures admit a Markovian coding.
