A linear symplectic form $ \omega$ on a vector space $V$ induces a symplectic structure (also denoted $ \omega$) on $V$ via the canonical identification of $TV$ with $V \times V$. The symplectic group $Sp(V )$ then embeds naturally in $Aut(V,\omega)$ (symplectromorphism), and the Lie algebra $sp(V)$ identifies with the subalgebra of $\chi(V, \omega)$ (Locally Hamiltonian Vector Field) consisting of vector fields of the form $$X(v) = Av$$ for some $A \in sp(V)$.
I think $Aut(V,\omega)$ has elements other than $Sp(V)$. For an operator in $Sp(V)$, it acts on the manifold $V$ in a homogeneous way. But I don't know how to find such an example, perphas in 2-dimensional case.
Another question is about the vector field $X(v)$, which is claimed to be the Hamiltonian vector field corresponding to $$Q_A(v) = \frac{1}{2} \omega(Av, v) $$ I try to show that for vector field $Y(v)$ $$ \omega(X,Y(v))=dQ_A(Y(v)) $$ The RHS equals to $$ dQ_A(Y(v))=\frac{1}{2}\omega(AY(v), v)+\frac{1}{2}\omega(Av, Y(v)). $$ But for $A\in sp(V)$, I don't know how to characterize this property. Thus, I am stuck. Any advice on this problem will be greatly appreciated. Thanks in advance.
Take any function $f$ with compact support and the Hamiltonian vector field defined by $\omega(X_f,.)=df$. The flow $\phi_t$ of $X_f$ is a flow of symplectomorphisms. If $f$ is not constant, then there exists $R$ such that for $\|x\|\geq R$, $\phi_t(x)=x$, but there exists $r>0$ such that the restriction of $\phi_t$ to the ball $B(0,r)$ is not the identity.