In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of the Lagrangian one looks at the hybrid moduli space problem. Recently I have been trying to understand how one can obtain the gluing of the trajectories on these moduli spaces to prove that we have a chain map between the Floer and Morse complexes.
There are two types of broken trajectories that can happen, either coming from the Morse-type problem or the Floer-type problem as explained in Proposition $3.6$. Let's for now focus on the Morse one which is Proposition $3.9$ part a. The authors claim that the gluing here works as in classical Floer theory. However I am not being able to even see how we can define the pre-gluing. Let's take $u\in \mathcal{M}^{+}(q_1,x)$ and $\bar q\in W^{u}(q_0)\cap W^{s}(q_1)$ now I would like to define a map $\bar q\#_{\rho} u$ such that
$$\lim_{\rho\rightarrow\infty}\bar q\#_{\rho}u=u , \text{in} \hspace{2mm}C^{\infty}_{loc}([0,\infty[\times [0,1],T^*M)$$ there exists a sequence of $\rho_k\rightarrow \infty$ and $t_k\in ]-\infty,0]$ such that $\phi^{t_k}_{-\nabla_{g,\mathcal{E}}}(\pi (\bar q\#_{\rho_k}u)(0,t))\rightarrow \bar q$ and $\pi(\bar q\#_{\rho}u(0,t))\in W^{u}(q_0)$.
However I am not being able to construct such a map. An idea I had was to use the conditions to create a path $\gamma_{\rho}:[0,\frac{1}{\rho}]\rightarrow M$ such that $\gamma_{\rho}(0)=\bar q$ and $\gamma_{\rho}(\frac{1}{\rho})=\pi(u(0,t))$. Then define
\begin{cases} (\bar q(t),p(t)) & s=0\\ (\gamma_{\rho}(t),p(t))& 0<s<\frac{1}{\rho}\\ u(s-\frac{1}{\rho}) & s\geq \frac{1}{\rho} \end{cases} where $u(0,t)=(\pi(u(0,t)),p(t))$. However this will not have the right properties since for the compact set $\{0\}\times [0,1]$ we have $\bar q\#_{\rho}u(0,t)=(\bar q(t),p(t))$ which does not converge to $u(0,t)$ as needed.
I have tried a couple of other things but none of them seemed to work and I am runnig out of ideas. Does anyone have any adivce on how I should do the pre-gluing in this case , or if even I need this kinda of approach to solve the problem ?
Any insight is appreciated, thanks in advance.
