, Volume 16, Issue 1, pp 47-63

The Moore-Myhill pseudo tiling for the Heisenberg tessellation automata

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Abstract

The notions of a Garden-of-Eden (GOE) configuration and mutually erasable configurations were discovered and investigated in the framework of Euclidean tessellations in the 1960s and 1970s. In 1993, Machi and Mignosi extended the GOE theorem for tessellation automata on Cayley graphs of non-exponential growth. We give an alternative proof to the GOE theorem for the discrete Heisenberg groups. Our proof has the advantage that it fully exploits the particular group structure so that a Moore-Myhill like tiling can be explicitly constructed. The tiling, as the wording suggests, is a spatially economic covering. The non-uniform packing method is introduced as a key technique for the construction.