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.