Question

This question is motivated by the classical fact from differential geometry :

Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ acts transtively on the configuration space of $n$-points in $M$ or equivalently it acts $n$-transitively on $M$.

As I recall, it is known that the symplectomorphism group $(M,\omega)$ acts transitively on $M$, which is assumed to be symplectic. My question is then the following :

Let $(M,\omega)$ be a symplectic manifold.
(i) When does $\textrm{Symp}(M,\omega)$ act $n$-transtively for $n\geq 2$ ?
(ii) If the answer above is NOT ALWAYS then what is known ?

As some background, the usual way one proves (rather the only way I know how to prove this) the first fact is by showing the following :
(i) for two sets of distinct $n$ points in $M$ given by $\{p_1,\ldots,p_n\}$ and $\{q_1,\ldots,q_n\}$ which are close, we find disjoint disks $D_i$'s containing $p_i,q_i$. This requires dimension at least $2$. Use some diffeomorphism of $D_i$ that is smoothly identity at the boundary and looks like a rotation inside $D_i$ that swaps $p_i$ and $q_i$.
(ii) Define the natural equivalence relation on $n$-tuples and observe that the configuration space of $n$-points in $M$. By (ii) each equivalence class is open. It is alsoclosed being the complement of open sets. Since the configuration space is connected (this requires dimension at least $2$) this means there is only one equivalence class.

Does this idea work in the symplectic setting - perhaps by taking paths $\gamma_i$ from $p_i$ to $q_i$ and getting Hamiltonian vector fields via $\omega(\gamma_i',\cdot)$ ?

Answer 1

Dear Somnath Basu, the answer to question is that, for any $k\geq 2$, the k-fold transitivity of the action of $\mathrm{Sympl}(M,\omega)$ on $M$ has only one obstruction, the trivial one, i.e. connectivity of $M$.
But has you proposed there is even more.

In particular Theorem A in a paper of W. Boothby says that,

given a connected symplectic manifold $(M,\omega)$, for any two sets $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ of disjoint point in $M$, where $n$ is arbitrary natural number, there exists a time dependent hamiltonian vector field $X_t$ of $(M,\omega)$ such that its evolution operator $K^X_{1,0}$ maps $x_i$ to $y_i$ for $i=1,\ldots,n$.

Answer 2

Very late, let me point out the following paper:

Peter W. Michor, Cornelia Vizman: n-transitivity of certain diffeomorphism groups. Acta Math. Univ. Comenianiae 63, 2 (1994),221--225. arXiv:dg-ga/9406005 (pdf)

There $n$-transitivity is proved for many groups of diffeomorphisms, in particular also for the groups of real analytic symplectic, or volume preserving, or contact diffeomorphisms.

Answer 3

For any finite number of points, the subgroup of Hamiltonian symplectomorphism that fixes some neighbourhood of the points acts transitively on the complement of the points.

Choose two points $a, b$ in the complement, and a path avoiding the constrained points joining them. Then you can find a (non-autonomous) Hamiltonian function with support in an arbitrarily small neighborhood of the path whose time-1 flow maps $a$ to $b$. Indeed, the group of Hamiltonian tranformations with support in a connected neighbourhood is transitive on Darboux balls charts, so the orbit of a point is open and closed.

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What this argument seems to prove is in fact the statement :

For any connected open subset $\Omega$ in a symplectic manifold, the group of Hamiltonian symplectomorphisms with support in $\Omega$ is transitive in $\Omega$.

In particular, the group of compactly Hamiltonian symplectomorphisms is $n$-transitive $\forall n$ on any connected symplectic manifold.