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Finite symmetry group actions on substitution tiling C*-algebras

For a finite symmetry group $G$ of an aperiodic substitution tiling system $(\p,ω)$, we show that the crossed product of the tiling C*-algebra $\Aw$ by $G$ has real rank zero, tracial rank one, a unique trace, and that order on its K-theory is determined by the trace. We also show that the action of $G$ on $\Aw$ satisfies the weak Rokhlin property, and that it also satisfies the tracial Rokhlin property provided that $\Aw$ has tracial rank zero. In the course of proving the latter we show that $\Aw$ is finitely generated. We also provide a link between $\Aw$ and the AF algebra Connes associated to the Penrose tilings.