I'm trying to get some intuition for the Conley-Zehnder index in the setting of Floer homology of a symplectomorphism $\phi : (M,\omega) \to (M,\omega)$. Let's assume that $\phi$ only has non-degenerate fixed points.
According to the paper of Dostouglu and Salamon "Self-Dual instants and Holomorphic curves" (DOI link pointing to publisher), the Maslov class determines a map $$ \mu : \text{Fix}(\phi) \to \Bbb Z_{2N}$$ (defined up to an additive constant) which satisfies $$\mu(u) = \mu_{CZ}(x_0) -\mu(x_1) \pmod{2N}$$ where $u$ is a Floer trajectory connecting $x_0$ to $x_1$ with Fredholm index $\mu(x)$.
I believe that they use this relation to define $\mu_{CZ}(x_1)$ say as $$\mu_{CZ}(x_1)=\mu_{CZ}(x_0) + \mu(u) \pmod{2N}$$ where the indeterminacy modulo $2N$ is given by the fact that by attaching a $J$-holomorphic sphere to the trajectory I can increase its Maslov index by $2N$ while preserving the fact that it's a solution of the Cauchy-Riemann equations. The indeterminacy up to translation comes from the fact that we are implicitly setting a value for index for $\mu_{CZ}(x_0)$, and we retrieve all the others starting from it.
Is this correct?
Why are these the only indeterminacies? I.e. why is $\mu_{CZ}(x_1)$ a well-defined number modulo $2N$ and not modulo some (smaller) divisors of $2N$? Equivalently, how can I see that the Fredholm index of a Floer trajectory going from $x_0$ to itself must be a multiple of $2N$?
If we were to prove that, as in the Hamiltonian case, the CZ-index of a critical point depends only depends on a choice of its lift to the Novikov cover of the twisted loop space of $\phi$ and that the Deck transformation group of such cover acts by attaching spheres like in the Hamiltonian case then we would be done I believe. But I can't see it stated anywhere and I'm not exactly sure it's the case. In fact if we consider the torus $T^2$, if the Deck transformation group of the Novikov cover acts by attaching spheres, then it would act trivially since $\pi_2(T^2)=0$. But in many papers dealing with surface automorphisms, the authors assume that the genus is at least $2$ to have an exact action functional (See my question here for example ). Hence I must confused somewhere along the way.
