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Description:
By the Shadowing Lemma we can shadow any sufficient accurate pseudo-trajectory of a hyperbolic system by a true trajectory of a hyperbolic system. If we are interested in finite trajectories, at least from one side, then a pseudo trajectory usually has many possible shadows. Here we show that we can choose a continuous single-valued selector from the corresponding multi-valued operator "pseudo-trajectory ↦ the totality of possible shadows". We do this in the context of Lipschitz mappings which are semi-hyperbolic on some compact subset, which need not be invariant. We also prove that semi-hyperbolicity implis inverse shadowing with respect to a very broad class of nonsmooth perturbations.