Given a Weinstein domain $M$ and a compactly supported, exact symplectomorphism $\phi$, one can construct \textbf{the open symplectic mapping torus} $T_\phi$. Its contact boundary is independent of $\phi$ and thus $T_\phi$ gives a Weinstein filling of $T_0\times M$, where $T_0$ is the punctured 2-torus. In this talk, we will outline a method to distinguish $T_\phi$ from $T_0\times M$ using dynamics and deformation theory of their wrapped Fukaya categories. This will involve the intermediate step of constructing a mirror symmetry inspired algebro-geometric model related to Tate curve for the wrapped Fukaya category of $T_\phi$ and exploiting the dynamics of these models to distinguish them.