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The centralizer of a $$C^1$$-generic diffeomorphism is trivial. (English) Zbl 1149.37012
This 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.
(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 $$\text{Diff}^r(M)$$?
For the case $$r=1$$ the authors give the following answer: For any compact, connected manifold $$M$$, there is a residual subset of $$\text{Diff}^1(M)$$ consisting of diffeomorphisms with trivial centralizer. This set does not contain any open and dense subset.
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.

##### MSC:
 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
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