This talk is about producing invariants to distinguish Liouville domains via the continuous dynamics on their wrapped Fukaya categories. Namely, given a Weinstein domain $M$ and a compactly supported symplectomorphism $φ$, one can construct a Weinstein domain called \textbf{the open symplectic mapping torus} and denoted by $T_φ$. The contact boundary of this manifold is independent of $φ$ and it is the same as the contact boundary of $T_0\times M$, where $T_0$ is the punctured torus. In this talk, we will show how to distinguish $T_φ$ from $T_0\times M$ via categorical dynamics. This involves the construction of a mirror symmetry inspired algebro-geometric model for the wrapped Fukaya category of $T_φ$ that is related to Tate curve and the (heuristic) use of ``rigid analytic'' dynamics of this model.

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