Extended symmetry groups of multidimensional subshifts with hierarchical structure

Abstract

The centralizer (automorphism group) and normalizer (extended symmetry group) of the shift action inside the group of self-homeomorphisms are studied, in the context of certain Z(d) subshifts with a hierarchical supertile structure, such as bijective substitutive subshifts and the Robinson tiling. Restrictions on these groups via geometrical considerations are used to characterize explicitly their structure: nontrivial extended symmetries can always be described via relabeling maps and rigid transformations of the Euclidean plane permuting the coordinate axes. The techniques used also carry over to the well-known Robinson tiling, both in its minimal and non-minimal versions.