On the barcode entropy of Lagrangian submanifolds

This article deals with relative barcode entropy, a notion recently introduced by Çineli, Ginzburg, and Gürel. It is a non-negative number assigned to a Hamiltonian diffeomorphism and a pair of Lagrangian submanifolds, and defined through Lagrangian Floer complexes associated to them. We exhibit several situations in which the relative barcode entropy 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 a certain set of periodic orbits in $ K $. Furthermore, we define a variant of relative barcode entropy and present first examples for which it is positive. As applications, certain robustness features of the volume growth and the topological entropy are discussed.