Symmetries and reversing symmetries of trace maps

TL;DR

This paper characterizes the reversing symmetry groups of 3D trace maps derived from substitution rules, revealing their structure as subgroups of polynomial mappings preserving a specific invariant, and linking them to the projective linear group.

Contribution

It provides a complete classification of reversing symmetry groups for trace maps associated with substitution rules, connecting dynamical symmetries to algebraic group structures.

Findings

Reversing symmetry groups are subgroups of polynomial mappings preserving I(x,y,z)

Trace maps form a group isomorphic to PGL(2,Z)

Complete characterization of symmetry groups for these dynamical systems

Abstract

A (discrete) dynamical system may have various symmetries and reversing symmetries, which together form its so-called reversing symmetry group. We study the set of 3D trace maps (obtained from two-letter substitution rules) which preserve the Fricke-Vogt invariant I(x,y,z). This set of dynamical systems forms a group G isomorphic with the projective linear (or modular) group PGL(2,Z). For such trace maps, we give a complete characterization of the reversing symmetry group as a subgroup of the group A of all polynomial mappings that preserve I(x,y,z).