Invariant sets for homeomorphisms of hyperbolic 3‐manifolds

Abstract We prove that under some assumptions on how points escape to infinity in the universal cover, homeomorphisms of hyperbolic 3‐manifolds are forced to have several invariant sets (in particular, they cannot be minimal). For this, we use some shadowing techniques which, when the homeomorphism has positive speed with respect to a uniform foliation, allow us to obtain strong consequences on the structure of the invariant sets. We discuss also homological rotation sets and end the paper with some extensions to other manifolds as well as posing some general problems for the understanding of minimal homeomorphisms of 3‐manifolds.