Reversing and extended symmetries of shift spaces

Michael Baake; John A. G. Roberts; Reem Yassawi

doi:10.3934/dcds.2018036

Article Contents

2018, Volume 38, Issue 2: 835-866. Doi: 10.3934/dcds.2018036

This issue Previous Article A Liouville-type theorem for cooperative parabolic systems Next Article A robustly transitive diffeomorphism of Kan's type

Reversing and extended symmetries of shift spaces

1.
Faculty of Mathematics, Universität Bielefeld, Box 100131,33501 Bielefeld, Germany
2.
School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia
3.
IRIF, Université Paris-Diderot — Paris 7, Case 7014,75205 Paris Cedex 13, France

Received: December 31, 2016

Revised: August 31, 2017

Published: November 2017

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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 $\mathbb{Z}^{d}$-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.

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Mathematics Subject Classification: 37B10, 37B50, 52C23, 20B27, 20H05.

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Figure 1. The chair inflation rule (upper left panel; rotated tiles are inflated to rotated patches), a legal patch with full $D_{4}$ symmetry (lower left) and a level-$3$ inflation patch generated from this legal seed (shaded; right panel). Note that this patch still has the full $D_{4}$ point symmetry (with respect to its centre), as will the infinite inflation tiling fixed point emerging from it

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Figure 2. The chair tiling seed of Figure 1, now turned into a patch of its symbolic representation via the recoding of Eq. (15). The relation between the purely geometric point symmetries in the tiling picture and the corresponding combinations of point symmetries and LEMs can be seen from this seed

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Figure 3. Illustration of the central configurational patch for Ledrappier's shift condition, which explains the relevance of the triangular lattice. Eq. (16) must be satisfied for the three vertices of all elementary $L$-triangles (shaded). The overall pattern of these triangles is preserved by all (extended) symmetries. The group $D^{}_{3}$ from Theorem 7 can now be viewed as the colour-preserving symmetry group of the 'distorted' hexagon as indicated around the origin

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