The talk is based on joint work with Jan BoroĊ„ski.

The aim of this talk is to present some open problems related to dynamics of homeomorphisms on hereditarily indecomposable continua (entropy, periodic points, etc.). We are going to present some partial answers in the literature, that may be of some help when searching for solutions to presented questions. Among other results, we are going to present a simple proof that when a map f on a topological graph induces a hereditarily indecomposable inverse limit with f as a single bonding map, then the entropy of induced shift homeomorphism is either zero or infinity.