Inverse shadowing and related measures

We study various weaker forms of the inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called ergodic inverse shadowing property (Birkhoff averages of continuous functions along an exact trajectory and the approximating one are close). We demonstrate that this property implies the continuity of the set of invariant measures in the Hausdorff metric. We show that the class of systems with ergodic inverse shadowing is quite broad; it includes all diffeomorphisms with hyperbolic nonwandering sets. Second, we study the so-called individual inverse shadowing (any exact trajectory can be traced by approximate ones, but this shadowing is not uniform with respect to the initial point of the trajectory). We demonstrate that this property is closely related to structural stability and Ω-stability of diffeomorphisms.