Abstract
We provide a proof of the Alpern multi-tower theorem for ℤ d actions. We reformulate the theorem as a problem of measurably tiling orbits of a ℤ d action by a collection of rectangles whose corresponding sides have no non-trivial common divisors. We associate to such a collection of rectangles a special family of generalized domino tilings. We then identify an intrinsic dynamic property of these tilings, viewed as symbolic dynamical systems, which allows for a multi-tower decomposition.
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