For Z^d substitution subshifts, the paper gives conditions for radius-zero extended symmetries to shuffle supertiles, proves the height lattice is invariant, and describes irregular fibres, but the proofs have significant gaps.
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Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts
For Z^d substitution subshifts, the paper gives conditions for radius-zero extended symmetries to shuffle supertiles, proves the height lattice is invariant, and describes irregular fibres, but the proofs have significant gaps.