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Characterization of planar pseudo-self-similar tilings. (English) Zbl 0997.52012
A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a non-exact substitution rule: the substitution for a tile is not geometrically similar to itself. An example is the Penrose tiling drawn with ‘thick’ and ‘thin’ rhombi.
The authors prove that a nonperiodic repetitive tiling of the plane is pseudo-self-similar if and only if it has a finite number of derived Voronoi tilings up to similarity. To establish this characterization, the authors settle a conjecture of E. A. Robinson by providing an algorithm which converts any pseudo-self-similar planar tiling into a self-similar tiling with some useful topological properties.

##### MSC:
 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 37A30 Ergodic theorems, spectral theory, Markov operators 37B15 Dynamical aspects of cellular automata 68R10 Graph theory (including graph drawing) in computer science
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