Inverse shadowing in actions of Baumslag—Solitar group

Abstract

In parallel with the shadowing theory (which is now very well-developed), the theory of inverse shadowing was developed. 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 the 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 generalise 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.