Reducing conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps

TL;DR

This paper presents a method to determine conjugacy of diffeomorphisms of the real line by reducing the problem to conjugacy within the subgroup of orientation-preserving maps, especially for degree -1 maps, using formal power series and existing centralizer results.

Contribution

It provides an explicit criterion to reduce conjugacy problems in the full diffeomorphism group to the subgroup of orientation-preserving maps, particularly for maps of degree -1.

Findings

Reduces conjugacy problem to subgroup for degree -1 maps

Explicit criterion involving squares of diffeomorphisms

Utilizes formal power series and Kopell's results

Abstract

Let $\Diffeo=\Diffeo(\R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\R$, under the operation of composition, and let $\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\in \Diffeo$ are conjugate in $\Diffeo$ to associated conjugacy problems in the subgroup $\Diffeo^+$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in $\Diffeo^+$, in order to ensure that $f$ is conjugated to $g$ by an element of $\Diffeo^+$. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a half-open interval.