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$$C^1$$-generic conservative diffeomorphisms have trivial centralizer. (English) Zbl 1149.37017
This paper deals with the problem to examine the centralizer of a diffeomorphism. Here, given a $$C^r$$-diffeomorphism $$f\in\text{Diff}^r(M)$$ the centralizer is defined as the set $$Z(f)=\{g\in \text{Diff}^r(M): fg=gf\}$$. The centralizer is trivial if $$Z(f)$$ only contains powers of $$f$$. The following question of S. Smale [Math. Intell. 20, No. 2, 7–15 (1998; Zbl 0947.01011)] serves as motivation: Consider the set of $$C^r$$-diffeomorphisms of a compact connected manifold $$M$$ with trivial centralizer. Is the set residual in $$\text{Diff}^r(M)$$?
This paper gives an affirmative answer to the above question in the case of symplectic and volume preserving diffeomorphisms. Specifically, let $$\text{Symp}^1(M)$$ denote the set of $$C^1$$-symplectomorphisms of $$M$$. Provided $$M$$ carries a volume $$\mu$$, then denote the set of $$C^1$$-diffeomorphisms of $$M$$ preserving $$\mu$$ by $$\text{Diff}^1_{\mu}(M)$$. The authors show that if $$M$$ is a compact, connected manifold of dimension at least $$2$$, then (a) the set of diffeomorphisms in $$\text{Symp}^1(M)$$ with trivial centralizers is a residual set, and (b) if $$\mu$$ is a volume on $$M$$, then the set of diffeomorphisms in $$\text{Diff}^1_{\mu}(M)$$ with trivial centralizer is a residual subset.

##### MSC:
 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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