Is it possible to approximate an area-preserving diffeomorphism $T$ of the disk $\mathbb{D}^2$ by a sequence of conjugates of periodic rotations $B_n^{-1} S_{\frac{p_n}{q_n}} B_n$, where $ S_{\frac{p_n}{q_n}} $ is the disk rotation of angle $2\pi \frac{p_n}{q_n}$ and $B_n$ is an area-preserving diffeomorphism of the disk?
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$\begingroup$ Do you know if this is true for $Diff^+(S^1)$? It's false in the subgroup $PSL_2(\mathbb{R})<Diff^+(S^1)$: the hyperbolic and elliptic elements are separated by the parabolic elements. So can you explain how a hyperbolic element in $PSL_2(\mathbb{R})$ is approximated by conjugates of elliptic elements in $Diff^+(S^1)$? I think the elements of non-zero rotation number in $Diff^+(S^1)$ are approximated by conjugates of rotations though (in fact, many are smoothly conjugate to a rotation). en.wikipedia.org/wiki/Denjoy's_theorem_on_rotation_number $\endgroup$– Ian AgolCommented Oct 8, 2013 at 5:11
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$\begingroup$ In which topology on the set of area-preserving diffeomorphisms do you wish to approximate the map $T$? $\endgroup$– Ian MorrisCommented Oct 8, 2013 at 13:42
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I think the answer in general is no and periodic points for $T$ are the main obstructions.
In fact, one can construct an area-preserving diffeomorphism $T$ with at least two hyperbolic fixed point. If the answer, in general, were "yes", one could approximate $T$ by diffeomorphisms which are conjugate to irrational rotations of $\mathbb{D}$, and henceforth, exhibiting only one fixed point.
But this is impossible because hyperbolic fixed points are "robust". More precisely, the amount of hyperbolic fixed points is locally constant when $\mathrm{Diff}^r(\mathbb{D})$ is endowed with the $C^r$-topology and $r\geq 1$. For the $C^0$-topology it is not true that the amount of fixed points is locally constant, but using an argument of indexes of curves and Lefschetz fixed point theorem, it can be easily shown that any homeomorphism in any sufficiently small $C^0$-neighborhood of $T$ has at least two fixed points, too.
On the other hand, there is a recent result due to Barney Bramham http://arxiv.org/abs/1204.4694 showing that the answer is yes when $T$ has only one periodic point.
