A Three-Dimensional SFT with Sparse Columns
Provides new examples of 3D subshifts with constrained subdynamics, relevant for symbolic dynamics and cellular automata theory.
The authors construct a nontrivial 3D subshift of finite type with sparse columns (at most two nonzero symbols per column) and show it is conjugate to spacetime diagrams of a partial cellular automaton. Variants with Wang cubes and binary alphabet are also presented.
We construct a nontrivial three-dimensional subshift of finite type whose projective $\Z$-subdynamics, or $\Z$-trace, is 2-sparse, meaning that there are at most two nonzero symbols in any vertical column. The subshift is deterministic in the direction of the subdynamics, so it is topologically conjugate to the set of spacetime diagrams of a partial cellular automaton. We also present a variant of the subshift that is defined by Wang cubes, and one whose alphabet is binary.