Conjugacy of real diffeomorphisms. A survey

TL;DR

This survey reviews classical and recent results on the conjugacy problem in the group of all diffeomorphisms of an interval, providing a systematic overview and new insights into the classification of conjugacy classes.

Contribution

It offers a comprehensive review of the conjugacy problem in Diffeo(I) and introduces new results for the general case, filling gaps in existing literature.

Findings

Classical solutions for special classes of diffeomorphisms

Identification of gaps in the literature

New results on conjugacy classification in the general case

Abstract

Given a group G, the conjugacy problem in G is the problem of giving an effective procedure for determining whether or not two given elements f, g of G are conjugate, i.e. whether there exists h belonging to G with fh = hg. This paper is about the conjugacy problem in the group Diffeo(I) of all diffeomorphisms of an interval I in R. There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.