Characterization of planar pseudo-self-similar tilings

A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a nonexact substitution rule: the substitution tor a tile is not geometrically similar to itself. An example is the Penrose tiling drawn with rhombi. We prove that a nonperiodic repetitive tiling of the plane is pseudo-selt-similar if and only if it has a finite number of derived Voronoi tilings up to similarity. To establish this characterization, we settle (in the planar case) a conjecture of E. A. Robinson by providing an algorithm which converts any pseudo-self-similar tiling of ℝ² into a self-similar tiling of ℝ² in such a way that the translation dynamics associated to the two tilings are topologically conjugate.