[Paper Review] Computation of extended symmetry groups for multidimensional subshifts with hierarchical structure

Álvaro Parra|arXiv (Cornell University)|Oct 5, 2018

Cellular Automata and Applications11 references4 citations

TL;DR

This paper introduces geometric techniques to analyze extended symmetry groups of multidimensional subshifts with hierarchical structure, such as bijective substitutive subshifts and the Robinson tiling. By examining large-scale patterns and fracture directions in special points, it proves that the extended symmetry group is virtually ℤᵈ and explicitly characterizes nontrivial symmetries as rigid transformations of the coordinate axes.

ABSTRACT

Automorphism groups are intrincate conjugacy invariants for subshifts, which can reveal important features of the dynamical structure of a shift action. One important case is the study of automorphism groups when the underlying subshift has a very rigid structure, e.g. substitutive subshifts or aperiodic constructions with large-scale self-similarity, such as the Robinson shift. In this work we study the automorphism group of bijective substitutive subshifts, and a potential generalization in the form of the group of extended symmetries, studied previously by Michael Baake, John Roberts and Reem Yassawi (arXiv:1611.05756); these symmetries, by allowing for shearing and other deformations of the underlying group, may reveal additional information of a geometric nature about the structure of these subshifts.

Motivation & Objective

• To understand the structure of extended symmetry groups in multidimensional subshifts with hierarchical organization.
• To identify constraints on symmetries by analyzing large-scale geometric features in special subshift points.
• To characterize the quotient group Sym(X,ℤᵈ)/Aut(X,ℤᵈ) as a set of rigid transformations of the coordinate axes.
• To extend techniques from the Robinson tiling to both minimal and non-minimal versions of the tiling.
• To establish that the extended symmetry group is virtually-ℤᵈ for bijective substitutive subshifts and the Robinson tiling.

Proposed method

• Uses the concept of fracture normal directions to identify forbidden symmetry directions in subshifts.
• Applies the Curtis-Hedlund-Lyndon theorem to represent sliding block codes via local functions and finite windows.
• Analyzes periodic points and their images under the maximal equicontinuous factor (MEF) to constrain fiber cardinality.
• Leverages the faithfulness of the shift action to ensure that symmetries preserve structural and dynamical properties.
• Employs recursive substitution structure (e.g., θᵐ(a)) to show that patterns with exponentially growing supports must align under symmetry.
• Uses contradiction arguments based on pattern overlaps and nontrivial intersections with fracture sets to rule out non-axis-parallel symmetries.

Experimental results

Research questions

• RQ1 What are the possible symmetries of a multidimensional subshift with hierarchical structure, such as a bijective substitutive subshift?
• RQ2 How can large-scale geometric features in special points of the subshift restrict the extended symmetry group?
• RQ3 What is the structure of the quotient group Sym(X,ℤᵈ)/Aut(X,ℤᵈ), and how can it be represented explicitly?
• RQ4 Can the techniques used for the minimal Robinson tiling be generalized to its non-minimal version?
• RQ5 To what extent do extended symmetries preserve the fiber structure of the maximal equicontinuous factor (MEF) in such subshifts?

Key findings

• The extended symmetry group Sym(X,ℤᵈ) is virtually-ℤᵈ for bijective substitutive subshifts and the Robinson tiling.
• Nontrivial extended symmetries are precisely those corresponding to rigid transformations of the coordinate axes, as elements of the quotient group Sym(X,ℤᵈ)/Aut(X,ℤᵈ).
• Fracture normal directions are restricted to the standard coordinate axes; no additional directions (e.g., diagonal) can support nontrivial symmetries.
• The proof relies on contradiction: assuming a non-axis-parallel fracture direction leads to a contradiction with pattern consistency across exponentially growing supports.
• The maximal equicontinuous factor (MEF) preserves fiber cardinality under extended symmetries, which restricts the possible mappings on periodic points.
• Techniques developed for the minimal substitutive subshift extend directly to the non-minimal version of the Robinson tiling, preserving symmetry constraints.