Reversibility in the diffeomorphism group of the real line

TL;DR

This paper characterizes reversible elements within the group of real-line diffeomorphisms, focusing on both the entire group and the subgroup of order-preserving transformations, providing a detailed understanding of their conjugacy properties.

Contribution

It offers a complete characterization of reversible elements in the diffeomorphism group of the real line, including the order-preserving subgroup, advancing the understanding of their algebraic structure.

Findings

Reversible elements are explicitly characterized in the full diffeomorphism group.

Reversible elements in the order-preserving subgroup are also fully described.

The conjugacy relations for reversibility are established for these groups.

Abstract

An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.