Given a path of symplectic matrices starting at the identity, the Conley-Zehnder index is essentially defined as twice the winding number plus some "correction" term if the path is not a closed path, but I do not understand where is the correction term coming from.
For example, if one computes the Reeb flow on the boundary of an ellipsoid $E(a,b,c)$ (assuming $a < b < c$, and the ratios of these radii are irrational) it acts as a rotation by $2\pi t/a$ in one plane, and rotations by angles $2\pi/b$ and $2\pi/c$ on the other planes. Let $\gamma$ denote the shortest orbit of period $a$. Choosing the trivialization from $E(a,b,c) \subset \mathbb{C}^3$ along $\gamma$, the linearization of the Reeb flow (along $\gamma$) yields a path $t\mapsto P(t)$ for $t \in [0,a]$, of symplectic matrices:
\begin{bmatrix}R_{2\pi t/a}&0&0\\0&R_{2\pi t/b}&0\\0&0&R_{2\pi t/c}\end{bmatrix}
and we can compute its CZ-index (i.e index of $\gamma$):
Since we achieve a full rotation on the first plane the index is just twice the winding number: 2
For the other two orbits, since $a < b < c$, the indices should be
$$CZ = 2 \times \text{Winding Number} + 1$$
Therefore, the index of $\gamma$ is
$$CZ(\gamma) = 2 + 1 + 1 = 4$$
Note we used the fact that $a/b$ and $a/c$ are less than one and irrational numbers.
I don't understand the "+1" in the computation of the indices for the second step. (Edited)
