A connectedness result for commuting diffeomorphisms of the interval

TL;DR

This paper proves that any two commuting group actions of Z^2 on the interval via higher smoothness diffeomorphisms can be continuously deformed into each other within a lower smoothness class, extending classical centralizer results.

Contribution

It establishes a connectedness result for representations of Z^2 in diffeomorphism groups, linking higher and lower smoothness classes and building on classical centralizer theorems.

Findings

Any two Z^2 representations in D^r_+[0,1], r >= 2, are connected by a path in D^1_+[0,1].

The result relies on classical theorems by Szekeres and Kopell on C^1 centralizers.

The work bridges higher and lower smoothness diffeomorphism groups for commuting actions.

Abstract

Let D^r_+[0,1], r >= 1, denote the group of orientation-preserving C^r diffeomorphisms of [0,1]. We show that any two representations of Z^2 in D^r_+[0,1], r >= 2, are connected by a continuous path of representations of Z^2 in D^1_+[0,1]. We derive this result from the classical works by G. Szekeres and N. Kopell on the C^1 centralizers of the diffeomorphisms of [0,1) which are at least C^2 and fix only 0.