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Let $(W,\omega)$ be a symplectic manifold and $H\in C^{\infty}(W\times S^{1};\mathbb{R})$. Let $(J_{t})_{t\in S^{1}}$ denote a family of $\omega$-compatible almost complex structures.

Fix a map $u\in C^{\infty}(\mathbb{R}\times S^{1};W)$ satisfying Floer's equation

$\partial_{s}u+J_{t}(u)(\partial_{t}x-X_{H}(u,t))=0.$

Assume that

$\lim_{s\rightarrow\pm\infty}u(s,t)=x^{\pm}(t),$

uniformly in $t$, where $x^{\pm}$ are two 1-periodic orbits of $H$.

Think of $u$ as a map $\mathbb{R}^{2}\rightarrow W$ that is 1-periodic in the $t$-direction. Recall that a regular point $(s,t)\in\mathbb{R}^{2}$ for a given map $u$ is a point such that:

(1) $\partial_{s}u(s,t)\ne0$

(2) $u(s,t)\ne x^{\pm}(t)$

(3) $u(s,t)\notin u(\mathbb{R}-s,t)$.

Floer, Hofer and Salamon proved that if $\partial_{s}u$ is not identically zero then the set $R(u)$ of regular points is open and dense in $\mathbb{R}^{2}$.

Suppose now that we are given a family $(H_{s})_{s\in\mathbb{R}}$ of functions such that $H_{s}=H^{+}$ for $s$ large and positive, and $H_{s}=H^{-}$ for $s$ large and negative. Assume also that $(J_{s,t})_{(s,t)\in\mathbb{R}\times S^{1}}$ also depends on $s$, but again is independent of $s$ for $\left|s\right|$ large. Consider again a map $u\in C^{\infty}(\mathbb{R}\times S^{1};W)$ satisfying

$\partial_{s}u+J_{s,t}(u)(\partial_{t}x-X_{H_{s}}(u,t))=0,$

$lim_{s\rightarrow\pm\infty}u(s,t)=x^{\pm}(t),$

uniformly in $t$, where now $x^{+}$ is a 1-periodic orbit of $H^{+}$, and $x^{-}$ is a 1-periodic orbit of $H^{-}$. We can still talk about regular points of $u$, which are defined in exactly the same way.

My question is: Is it known whether the set $R(u)$ of regular points of $u$ is still open and dense in $\mathbb{R}^{2}$ in this $s$-dependent case?

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