On the barcode entropy of Lagrangian submanifolds

This article deals with relative barcode entropy, a notion that was recently introduced by Cineli, Ginzburg, and Gurel. We exhibit some settings in closed symplectic manifolds for which the relative barcode entropy of a Hamiltonian diffeomorphism and a pair of Lagrangian submanifolds is positive. In analogy to a result in the absolute case by the above authors, we obtain that the topological entropy of any horseshoe K is a lower bound if the two Lagrangians contain a local unstable resp. stable manifold in K. In dimension 2, we also estimate the relative barcode entropy of a pair of closed curves that lie in special homotopy classes in the complement of certain periodic orbits in K. Furthermore, we define a variant of relative barcode entropy and exhibit first examples for which it is positive. As applications, certain robustness features of the volume growth and the topological entropy are discussed.