The centralizer of a \(C^1\)-generic diffeomorphism is trivial.

*(English)*Zbl 1149.37012This interesting and well written paper gives the answers on some questions of S. Smale, see [Math. Intell. 20, No. 2, 7–15 (1998; Zbl 0947.01011)]. More detailed, let us consider a compact, connected manifold \(M\) and consider the space \(\text{Diff}^r(M)\) of \(C^r\) diffeomorphisms of \(M\). The centralizer of \(f\in \text{Diff}^r(M)\) is defined as \(Z^r(f)=\{g\in \text{Diff}^r(M): fg=gf \}\). The question is: Consider the set of \(C^r\) diffeomorphisms from \(\text{Diff}^r(M)\) with trivial centralizer.

Let us note that knowing the centralizer of a diffeomorphism gives answer to more concrete questions as well, such as the embeddability of a diffeomorphism in a flow and the existence of roots of a diffeomorphism.

- (1)
- Is this set dense in \(\text{Diff}^r(M)\) ?
- (2)
- Is it residual in \(\text{Diff}^r(M)\) ?
- (3)
- Does it contain an open and dense subset of \(\text{Diff}^r(M)\)?

Let us note that knowing the centralizer of a diffeomorphism gives answer to more concrete questions as well, such as the embeddability of a diffeomorphism in a flow and the existence of roots of a diffeomorphism.

Reviewer: Alois Klíč (Praha)