Oriented Shadowing by Elementary Pseudotrajectories Near Attractors in a Banach Space

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.