Let $M$ be a compact manifold and let $\mathrm{Diff}^{1}(M)$ denote the group of $C^{1}$-diffeomorphisms of $M$. Let $Z(f) := \{g \in \mathrm{Diff}^{1}(M) \mid gf = fg \}$ denote the centralizer of $f$. Note that the cyclic subgroup $\langle f \rangle$ generated by $f$ is always in $Z(f)$. If $Z(f) = \langle f \rangle$, then we say that $f$ has trivial centralizer. Bonatti-Crovisier-Wilkinson showed in 2008 (arXiv, Publ. IHES 2009) that the group of $C^{1}$-diffeomorphisms of a compact manifold $M$ contains a residual subset of diffeomorphisms whose centralizers are trivial.
Suppose that $\dim M \geq 2$. Is anything analogous to this known in the more restrictive setting of $\mathrm{Diff}^{1}_{\mathrm{vol}}(M)$, the group of $C^{1}$-diffeomorphisms of a compact manifold preserving a volume form?
Any comments / references are greatly appreciated! Thanks!
Edit: I am primarily interested in the group $\mathrm{Diff}^{\infty}_{\mu}(\mathbb{S}^2)$ of smooth diffeomorphisms of the $2$-sphere preserving the standard area form $\mu$. I asked the above question, since it is more closely connected to the work of Bonatti-Crovisier-Wilkinson. Of couse, if anyone knows anything specifically pertaining to $\mathrm{Diff}^{\infty}_{\mu}(\mathbb{S}^2)$, that would be great!
