Inverse Shadowing in Actions of a Baumslag–Solitar Group

Abstract

In parallel with the shadowing theory (which is now very well-developed), the theory of inverse shadowing has been advanced. The main difference between the two theories is that the shadowing property means that we can find an exact trajectory near an approximate one while inverse shadowing means that, given a family of approximate trajectories, we can find a member of this family that is close to any chosen exact trajectory. We generalize the property of inverse shadowing for group actions and prove the absence of this property for some linear actions of the Baumslag–Solitar group, which is often considered as a source of counterexamples.