  In this paper, we present partial results towards a classification of
symplectic mapping tori using dynamical properties of wrapped Fukaya
categories. More precisely, we construct a symplectic manifold $T_\phi$
associated to a Weinstein domain $M$, and an exact, compactly supported
symplectomorphism $\phi$. $T_\phi$ is another Weinstein domain and its contact
boundary is independent of $\phi$. In this paper, we distinguish $\phi$ from
$T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we
obtain pairs of diffeomorphic Weinstein domains with the same contact boundary
and whose symplectic cohomology groups are the same, as vector spaces, but that
are different as Liouville domains. To our knowledge, this is the first example
of such pairs that can be distinguished by their wrapped Fukaya category.
  Previously, we have suggested a categorical model $M_\phi$ for the wrapped
Fukaya category $\mathcal{W}(T_\phi)$, and we have distinguished $M_\phi$ from
the mapping torus category of the identity. In this paper, we prove
$\mathcal{W}(T_\phi)$ and $M_\phi$ are derived equivalent (Theorem 1.9); hence,
deducing the promised Theorem 1.1. Theorem 1.9 is of independent interest as it
preludes an algebraic description of wrapped Fukaya categories of locally
trivial symplectic fibrations as twisted tensor products.
