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Let $X$ be a compact Riemann surface (one dimensional complex manifold) of genus $g > 1$, fix a non-degenerate two form $omega$ on $X$, it is automatically closed by dimension reasons, so it is a symplectic structure on $X$.

A symplectomorphisms of $(X,omega)$ is a diffeomorphisms $g$ of $X$ preserving form $omega$ i.e. $g^* omega = omega$.

What we can say about the group of all symplectomorphisms of $X$? Is it related to analytic automorphisms of $X$? Is it finite or infinite? What are orbits of the action on $X$? I don't know anything about this group, except the definition.

You can say an immense amount.

This is the simplest possible case of the groups $text{Symp}(M)$, where $M$ is a symplectic manifold. First allow me to say some generalities on those.

If $M$ is a connected symplectic manifold, the group $text{Symp}(M)$ acts transitively on ordered sequences $(x_1, cdots, x_n) in M^n$ so that all $x_i neq x_j$ (unless $i = j$). This is called "$n$-transitivity", and should be read as saying "I can take any $n$ distinct points to any other $n$ distinct points". This is easily implied by the claim that the group $text{Symp}_c(M)$ of compactly supported symplectomorphisms (those that are the identity outside of a compact set) acts transitively on $M$. I will leave this implication as an exercise to you, but explain why $text{Symp}_c$ acts transitively.

This is best discussed by the construction of Hamiltonian vector fields. Given a function $f$ on a domain, one defines $X_f$ to be the unique vector field with $omega(X_f, Y) = df(Y)$ for all vector fields $Y$. Then flow along $X_f$ preserves the symplectic form, and $text{supp}(X_f) subset text{supp}(f)$. Now it suffices to show that the orbits of $text{Symp}_c$ are open sets, as then we would have a decomposition of $M$ into disjoint open sets; thus orbits would be all of $M$. So we only need to construct a Hamiltonian vector field taking us to a nearby point.

So pick a standard symplectic chart around each $x in M$; we will show that in $U subset Bbb R^{2m}$, there is a Hamiltonian vector field, compactly supported in $U$, flowing $0$ to any point sufficiently close to $0$.

To do this, if $v in Bbb R^{2m}$ is any vector, let $f_v(x) = langle v, xrangle$; because this is linear, we also have $df_v = langle v, -rangle$ at every point. Because the symplectic form here is $omega(X_v, x) = langle X_v, Jxrangle = langle -JX_v, xrangle$, and by definition $omega(X_v, x) = df_v(x) = langle v, xrangle$, we find that we must have $X_v = Jv$, where $J$ is the multiplication-by-$i$ map on $Bbb R^{2m} = Bbb C^m$.

In particular, $X_v$ is just "flow in the constant direction $Jv$". Replacing $f_v$ with $beta f_v$, where $beta$ is a bump function equal to $1$ near $0$ and supported inside a compact subset of $U$, we obtain a corresponding vector field $X'_v$ which is compactly supported inside $U$, but for small $t$ the time-$t$ flow just sends $0$ to $tJv$, and hence an orbit of $text{Symp}_c(M)$ is open, as desired.

In particular, note that $text{Symp}_c(M)$ must be infinite-dimensional, given that it has a surjective map to an open subset of $M^n$ for all $n$ (that is, $n$-transitivity means this group must be gigantic).

In dimension $n = 1$ (real dimension 2), however, things become much more simple: $text{Symp}(M) = text{Vol}(M)$, the group of diffeomorphisms fixing a particular volume form. This group similarly acts $n$-transitively in any dimension, and is in many ways much simpler than the symplectomorphism group. In fact, the original Moser trick proves a few facts: 1) any two volume forms $omega_1, omega_2$ of the same volume are related by some diffeomorphism $f$; and 2) the full group $text{Diff}(M)$ is homotopy equivalent to the subgroup $text{Vol}(M)$ of diffeomorphisms preserving a fixed volume form. So there is "no new topology to be had" in studying volume-preserving diffeomorphisms. This is very much not true in the symplectic case (in higher dimensions, of course).

None of these are related to analyticity (real or complex).

For lots of interesting facts about many related groups (Diff, Symp, Vol, ...) see Banyaga's book on the subject. One famous fact: when $M$ is compact, the connected component of the identity $text{Vol}_0(M)$ is a simple group. (In the symplectic case, one has to further restrict to the kernel of a certain map called the flux homomorphism; see Banyaga's book or the shorter article by McDuff.)