Notations : Suppose V is a closed contact compact manifold with contact form $\alpha$, of dimension 2n+1. Consider the symplectic sub-bundle $ \xi \subset TV $ given by $ \xi=$ ker($\alpha$). So $ \xi \rightarrow V $ is a vector bundle of rank $2n$ admitting the symplectic structure $d\alpha$. Also consider the Reeb vector field "$R$" determined by $\alpha$ and corresponding flow $\phi\_t$. Assume that there is a finite number of simple periodic orbits of Reeb flow and all of them are non-degenerate.


Here is the question :
In general there are plenty of almost complex structure on the symplectic bundle $\xi$ but is there any one which is invariant under Reeb flow, $\phi\_t$ ?


I believe the answer will be no in general due to complicated behavior of $\phi\_t$, but 
I wish that I am not right.



Looking for almost complex structure on a contact manifold invariant under flow of Reeb vector field !? 
In general there is no invariant complex structure.


Let $\gamma$ be a closed orbit of the Reeb field. Consider a linearization $A$ of the Poincare return map along $\gamma$. $A$ is not, in general, a realification of a complex operator (with respect to any arbitrary complex structure). For example, as far as I remember, its Lefschetz number det(1−A) could be negative, which is impossible for a realification of a complex operator.


