1.4. CONTRUCTING INTERESTING TILINGS 19 Kite Dart Figure 1.16. Penrose kite and dart tiles, with bumps to enforce matching rules A remarkable theorem of Shahar Mozes [Moz] says that for any substi- tution σ involving square tiles of equal size, there exists a set S and a factor map f : ΩS Ωσ that is onto, measure-preserving and 1:1 away from a negligible set (meaning a set of measure zero with respect to all translation- invariant probability measures). Mozes’ argument was generalized by Radin [Rad] to apply to the pinwheel tiling, and Mozes’ theorem was generalized by Goodman-Strauss [GS] to apply to any substitution tiling (as defined in this book). Despite their historical significance, and despite Goodman-Strauss’ the- orem, we will not be talking much about matching rules tiling spaces. While powerful techniques have been developed to explore the topology of substi- tution and cut-and-project tiling spaces, at this time the topology of most local matching rules tiling spaces remains a mystery.
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