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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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