Abstract:
Let Φ be a uniformly continuous action of a finitely generated group G on a metric space.
The shadowing property of Φ means that, given an approximate trajectory, we can find an exact trajectory close to it. The inverse shadowing property of Φ means that, given a family of approximate trajectories (generated by a so-called approximate method), for any fixed exact trajectory of Φ, we can find a member of this family that is close to this fixed trajectory.
The Reductive Shadowing Theorem (RST) states that if the action of a one-dimensional subgroup of G is topologically Anosov (i.e., it has the shadowing property and is expansive), then the action Φ is topologically Anosov as well (and hence, Φ has the shadowing property).
The first RST was proved in [1] for the groups Zp ; later it was generalized to the case of virtually nilpotent groups [2]. At the same time, it was shown in [2] that the RST is not valid, for example, for the Baumslag–Solitar groups BS(1, n) with n > 1.
It is shown in [3] that an analog of the RST for the case of inverse shadowing (with “topologically Anosov” replaced by the so-called “Tube Condition”) is also valid for virtually nilpotent groups.
[1] S.Yu.Pilyugin and S.B.Tikhomirov, Shadowing in actions of some abelian groups, Fund. Math., 179 (2003), 83-96.
[2] A.V.Osipov and S.B.Tikhomirov, Shadowing for actions of some fifinitely generated groups, Dyn. Syst., 29 (2014), 337-351.
[3] S.Yu.Pilyugin, Inverse shadowing in group actions, Dyn. Syst., 32 (2017), 198-210.
