Rhombus Tilings: Decomposition and Space Structure

We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope. Two tilings are linked if one can pass from one to the other by a local transformation, called a flip. We first use a decomposition method to encode rhombus tilings and give a useful characterization for a sequence of bits to encode a tiling. We use the previous coding to get a canonical representation of tilings, and two order structures on the space of tilings. In codimension 2 we prove that the two order structures are equal. In larger codimensions we study the lexicographic case, and get an order regularity result.