Beta-expansions and multiple tilings

Abstract: We introduce a class of piecewise linear transformations that can be used to generate beta-expansions with arbitrary digits. Under some conditions on beta and the digit set, we can construct a natural extension for such a transformation, which allows us to get an invariant measure equivalent to Lebesgue for the original transformation. From the natural extension, we obtain a multiple tiling of a Euclidean space. For the classic greedy beta-transformation (Tx = beta x (mod 1)) the Pisot conjecture states that this construction gives a proper tiling for all Pisot numbers beta. We give an example of a double tiling, showing that this conjecture is no longer true in the more general setting.