Reversing and extended symmetries of shift spaces

TL;DR

This paper explores the structure of reversing and extended symmetry groups in symbolic dynamics, providing explicit descriptions for certain well-known shift systems and developing a general theory for higher-dimensional cases.

Contribution

It introduces the concept of extended symmetries in higher-dimensional shift spaces and determines their groups for specific models like the chair tiling and Ledrappier's shift.

Findings

Explicit symmetry groups for Sturmian, Thue--Morse, and Rudin--Shapiro shifts.

Development of a general theory for extended symmetries in $\

determination of extended symmetry groups for the chair tiling and Ledrappier's shift.

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 \emph{extended symmetries}. We develop their basic theory for faithful $\mathbb{Z}^{d}$-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.