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.
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