Sur les rapprochements par conjugaison en dimension 1 et classe C^1

TL;DR

This paper demonstrates that the space of C^1 actions of Z^d on the interval or circle is connected, with all actions C^0 conjugate to trivial or rotation actions, and extends results to nilpotent groups.

Contribution

It establishes the connectedness of C^1 actions of Z^d on 1D manifolds and extends the framework to nilpotent group actions.

Findings

All C^1 actions can be C^0 conjugated into actions converging to trivial or rotation actions.

The space of such actions is connected via arcs.

Extensions are provided for nilpotent group actions.

Abstract

We prove that the space of actions of Z^d by C^1 (orientation-preserving) diffeomorphisms of either the interval or the circle is connected by arcs. This is proved by showing that all such actions can be C^0 conjugated via a 1-parameter family into diffeomorphisms that converge to either the trivial action or an action by Euclidean rotations. Extensions for nilpotent group actions are provided.