I'm looking for a clarification of a construction done in Seidel's Symplectic Floer Homology of a Dehn twist: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for writing the equations for fixed points Floer homology.
Let $\phi : \Sigma \to \Sigma$ be a symplectomorphism with non-degenerated fixed points. To define $HF_*(\phi)$, among many other things, we need to equip $\Sigma$ with a family of almost complex structures $J_t$ such that $J_{t+1}=\phi^*J_t$.
In the aforementioned paper, this is done in steps and the one that I don't understand is the following:
at page 832 the author claims that on the neighbourhood of the Lagrangian sphere $[-R,R]\times S^1$ there are standard almost complex structures $\{J_t^R\}$ that can be extended to the entire $\Sigma$
The problem that I have is that I don't see why $(T_0^R)^*J_t^R=J_{t+1}^R$ on the cylinder. For me the standard almost complex structure on $[-R,R]\times S^1$ is the (constant family) that in the canonical coordinates looks like $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ but in this case it's immediate to verify by hand that $$(T_0^R)^*J_t^R \neq J_{t+1}^R$$ It's very likely that I'm missing something, but I really don't know what, hence I decided to ask the question once and for all.
what are these standard almost complex structures on the cylinder that are suitable for setting up the Floer equations in this case?
