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 for 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-self-similar if and only if it has a finite number of derived Vorono\"{\i} 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 R ² into a self-similar tiling of R ² in such a way that the translation dynamics associated to the two tilings are topologically conjugate. Received June 20, 2000, and in revised form January 25, 2001. Online publication July 25, 2001.