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Extended symmetry groups of multidimensional subshifts with hierarchical structure

Banff International Research Station for Mathematical Innovation and Discovery

We discuss the automorphism group, i.e. the centralizer of the shift action inside the group of self-homeomorphisms of a subshift, together with the extended symmetry group (the corresponding normalizer) of certain $\mathbb Z^d$-subshifts with a hierarchical structure, like bijective substitutive subshifts and the Robinson tiling. This group has been previously studied in e.g. Michael Baake, John Roberts and Reem Yassawi's previous works, among others.

Treating those subshifts as geometrical objects, we introduce techniques to identify allowed extended symmetries from large-scale structures present in certain special points of the subshift, leading to strong restrictions on the group of extended symmetries. We prove that in the aforementioned cases, $\text{Sym}(X, \mathbb Z^d)$ (and thus $\text{Aut}(X, \mathbb Z^d)$) is virtually-$\mathbb Z^d$ and we explicitly represent the nontrivial extended symmetries, associated with the quotient $\text{Sym}(X, \mathbb Z^d)/\text{Aut}(X, \mathbb Z^d)$, as a subset of rigid transformations of the coordinate axes. We also show how our techniques carry over to the study of the Robinson tiling, both in its minimal and non-minimal version. We emphasize the geometric nature of these techniques and how they reflect the capability of extended symmetries to capture such properties in a subshift.

Mathematics; Dynamical systems and ergodic theory; Group theory and generalizations; Dynamical systems

video/mp4

Author affiliation: Universidad de Chile

2021-01-16

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77113

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/