The Z^d Alpern multi-tower theorem for rectangles: a tiling approach

TL;DR

This paper proves the Alpern multi-tower theorem for Z^d actions by reformulating it as a tiling problem with rectangles, introducing a new approach using generalized domino tilings and symbolic dynamics.

Contribution

It provides a novel tiling-based proof of the multi-tower theorem for higher-dimensional actions, connecting tiling properties with dynamical decompositions.

Findings

Establishes a tiling characterization of the multi-tower theorem.

Identifies an intrinsic dynamic property of domino tilings enabling the decomposition.

Links symbolic dynamical systems with multi-tower structures in Z^d actions.

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.