Abstract
We study shadowing properties of differentiable mappings in Banach spaces in neighborhoods of attractors. The properties of oriented shadowing of a pseudotrajectory are defined. A pseudotrajectory \(\xi=\{x_{k}:k\in{\mathbb{Z}}\}\) belonging to an attractor \({\mathcal{A}}\) of a mapping \(f\) is called elementary if the set of indices \({\mathbb{Z}}\) can be decomposed into a finite family of intervals \(I_{1},\dots,I_{n}\) so that for any set \(I_{m}\) there exist two fixed hyperbolic points \(p\) and \(q\) such that the set of points \(\{x_{k}:k\in I_{m}\}\) of the pseudotrajectory belongs to a trajectory lying in the intersection of the unstable manifold of the point \(p\) and the stable manifold of the point \(q\) . The main results of the paper state that if \(f\) is gradient-like with hyperbolic nonwandering set in \({\mathcal{A}}\) , then \(f\) has the properties of oriented shadowing by elementary pseudotrajectories belonging to \({\mathcal{A}}\) and to a neighborhood of \({\mathcal{A}}\) . As an application, we consider semigroups generated by parabolic PDEs.
