A brief guide to reversing and extended symmetries of dynamical systems

TL;DR

This paper reviews the concepts of reversing symmetries and their extensions in dynamical systems, highlighting recent developments and differences between nonlinear and symbolic dynamics, especially in higher-dimensional shifts.

Contribution

It provides an informal overview of reversing symmetry groups, their extensions, and recent advances, connecting classical and symbolic dynamical systems.

Findings

Reversing symmetry groups extend classical symmetry groups in dynamical systems.

Differences between nonlinear and symbolic dynamics are highlighted.

Recent developments include extensions to higher-dimensional shifts via automorphism groups.

Abstract

The reversing symmetry group is a well-studied extension of the symmetry group of a dynamical system, the latter being defined by the action of a single homeomorphism on a topological space. While it is traditionally considered in nonlinear dynamics, where the space is simple but the map is complicated, it has an interesting counterpart in symbolic dynamics, where the map is simple but the space is not. Moreover, there is an interesting extension to the case of higher-dimensional shifts, where a similar concept can be introduced via the centraliser and the normaliser of the acting group in the full automorphism group of the shift space. We recall the basic notions and review some of the known results, in a fairly informal manner, to give a first impression of the phenomena that can show up in the extension from the centraliser to the normaliser, with some emphasis on recent developments.