In this paper, on the one hand, we prove that for \(n \ge 2\) 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 and Xue (Nonlinearity 33:6297–6348, 2020) (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 and Xue (Nonlinearity 33:6297–6348, 2020), showing that any \({\widetilde{\pi }}_1(M)\) -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.
