  We study various weaker forms of inverse shadowing property for discrete
dynamical systems on a smooth compact manifold. First, we introduce the
so-called Ergodic Inverse Shadowing property (Birhhoff averages of continuous
functions along the exact trajectory and the approximating one are close). We
demonstrate that this property implies continuity of the set of invariant
measures in Hausdorff metrics. We show that the class of systems with Ergodic
Inverse Shadowing is quite broad, it includes all diffeomorphisms with
hyperbolic nonwandering sets. Secondly, 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 selection of the initial point of
the trajectory). We demonstrate that this property is closely related to
Structural Stability and $\Omega$-stability of diffeomorphisms.
