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A substitution tiling is a certain globally de ned hierarchical structure in a geometric space; we show that for any substitution tiling in E n , n 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, in nite collections of forced aperiodic tilings are constructed. On the left in gure 1, L-shaped tiles are repeatedly in ated and subdivided". We de ne our terms more precisely in Section 1. As this process is iterated, larger and larger regions of the plane are tiled with L-tiles hierarchically

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

No abstract

We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This can also be regarded as a "regular production system" [5] that does admit bi-infinite orbits, but admits no periodic orbits.