  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.
