Reversibility in Dynamics and Group Theory

Abstract

Reversibility is a thread woven through many branches of mathematics. It arises in dynamics, in systems that admit a time-reversal symmetry, and in group theory where the reversible group elements are those that are conjugate to their inverses. However, the lack of a lingua franca for discussing reversibility means that researchers who encounter the concept may be unaware of related work in other fields. This text is the first to make reversibility the focus of attention. The authors fix standard notation and terminology, establish the basic common principles, and illustrate the impact of reversibility in such diverse areas as group theory, differential and analytic geometry, number theory, complex analysis and approximation theory. As well as showing connections between different fields, the authors' viewpoint reveals many open questions, making this book ideal for graduate students and researchers. The exposition is accessible to readers at the advanced undergraduate level and above.

Reversing and extended symmetries of shift spaces

“…In general, however, it will be a true subgroup, because we do not allow for situations where S is conjugated into some other symmetry of infinite order, say. Nevertheless, for shifts (X, S) with faithful action, one has R(X) = norm Aut(X) S ; see [11,50] for more complicated situations.…”

“…Much later, this approach was re-analysed and extended in a more group-theoretic setting [44,31], via the introduction of the (topological) symmetry group S(X) of (X, T ) and its extension to the reversing symmetry group R(X). The latter, which is the group of all selfconjugacies and flip conjugacies of (X, T ), need not be mediated by an involution as in the original setting; see [10,9,47] for examples and [45,11,50] for some general and systematic results. Previous studies have often been restricted to concrete systems such as trace maps [53], toral automorphisms [10] or polynomial automorphisms of the plane [9]; see [50] for a comprehensive overview from an algebraic perspective.…”

A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems

“…As is implicit from our formulation so far, reversibility is not an interesting concept when T itself is an involution. More generally, when T has finite order, the structure of R(X, T ) is a group-theoretic problem, and of independent interest; see [61] for a concise exposition. However, in the context of dynamical systems, one is mainly interest in the case that T ≃ Z.…”

“…In this section, we will describe, in a somewhat informal manner, how symmetries and reversing symmetries arise in three particular families of dynamical systems, namely trace maps, toral automorphisms, and polynomial autormorphisms of the plane. Clearly, there are many other relevant examples, some of which can be found in [72,54,61] and references therein.…”

Number-theoretic positive entropy shifts with small centralizer and large normalizer

“…Here, we adopt the point of view of [2,9] to analyse both the (topological) centralizer (denoted by S below) and the normalizer of the shift space, the latter denoted by R, as this pair can be quite revealing as soon as d 2. In fact, both the topological setting and the extension to higher dimensions go beyond some of the initial studies [21,26] that specifically looked at reversibility in the measure-theoretic setting for d = 1; see [2,31,33] and references therein for more on the early reversibility results. Further, the groups S and R are often explicitly accessible, both for systems of low complexity, where S is often minimal due to some form of topological rigidity, and beyond, where other rigidity mechanisms of a more algebraic nature emerge.…”