Arc-connectedness for the space of smooth $\mathbb{Z}^d$-actions on 1-dimensional manifolds

Abstract

We show that the space of Z d actions by C 2 orientation-preserving diffeomorphisms of a compact 1-manifold is C 1+ac arcwise connected.Centralizers of diffeomorphisms can be viewed as infinitesimal symmetries of a given dynamical system. Starting with the seminal work of Nancy Kopell [10], they have become a central object of study in dynamics. Perhaps the most relevant recent work on this is [1], which solves a longstanding question of Stephen Smale about centralizers of generic diffeomorphisms and, also, is a wonderful "window" to enter into this huge subject. However, despite the effort of many people, several natural problems remain unsolved. Here we deal with a longstanding question raised in the seventies by Harold Rosenberg [15], that partly inspired the thesis of Jean-Christophe Yoccoz [22]: Is the space of (orientation-preserving) commuting circle diffeomorphisms locally arcwise connected ? Quite surprisingly, this has revealed as a very difficult question, and only a few (and somewhat recent) results in this direction are known. These may be summarized as follows (in all what follows, all maps in consideration are assumed to preserve the orientation):Remark 1.2. The proof above actually shows more: the natural inclusion of SO(2, R) d in the space of Z d -actions on the circle (endowed with the compact-open topology) is a homotopy equivalence. (Compare [7, Propositions 4.2].) We do not know whether this result extends to higher regularity.