Reversing and extended symmetries of shift spaces

Abstract

The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue-Morse system with various generalisations or the Rudin-Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of extended symmetries. We develop their basic theory for faithful Z d -actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier's shift, which is an example of algebraic origin.

Realization of big centralizers of minimal aperiodic actions on the Cantor set

“…In this case, none of the former restrictions appears: the centralizer can be uncountable as in the odometer action, or not residually finite as it was shown in [3], where the authors construct a minimal subshift whose centralizer contains a group isomorphic to the rationals Q. It appears from recent works that the centralizers of zero entropy minimal subshifts are very limited [2,7,8,9,10,11,12,13]. However the size of the centralizer does not restrict the complexity function.…”

A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems

“…Consequently, this example provides ample evidence that one also needs to consider the normaliser. The general result reads as follows; see [20] for the details.…”

“…Let us collect a few examples of reversible subshifts, in an informal manner; see [20] and references therein for precise statements and proofs, and [26,69,3,64,8] for general background on substitution generated subshifts. Among these examples are (1) the full shift [51,57], X = A Z , where S(X) is huge (and not amenable);…”

“…Its symmetry group is Z × C 2 , and it is reversible, but no reversor is an involution. Instead, there is a reversor of order 4 (and all reversors have this order), and the reversing symmetry group is Z ⋊ C 4 , where the square of the generating element of the cyclic group C 4 is the extra (involutory) symmetry; see [20] for details on this and the previous examples.…”

“…Unlike before, the structure of the normaliser is generally much richer now, which also means that R(X) is a considerably better topological invariant than S(X). Indeed, the normaliser can even be an infinite extension of the centraliser when d > 1, as can be seen from the full shift as follows; see [20,Lemma 4].…”