Notes on symplectic squeezing in $T^* \mathbb T^n$ and spectra of Finsler dynamics

In this paper, on the one hand, we prove that for $n \geq 2$ any subbundle of $T^* \mathbb T^n$ with bounded fibers symplectically embeds into a trivial subbundle of $T^* \mathbb T^n$ where the fiber is an irrational cylinder. This not only resolves an open problem in Gong-Xue's recent work (which was stated for the 4-dimension case, that is, $n =2$) and also generalizes to any higher-dimensional situation. The proof is based on some version of Dirichlet's approximation theorem. On the other hand, we generalize a main result in Gong-Xue's work mentioned above, showing that any topologically trivial Liouville diffeomorphism on $T^*M$ (for instance, a diffeomorphism induced by an isometry on $M$) does not change the full marked length spectrum of a Finsler metric $F$ on $M$, up to a lifting of the Finsler metric $F$ to the unit codisk bundle $D^*_FM$. The proof is based on persistence module theory.