The shadowing lemma

A dynamical system has the shadowing property if every pseudo-trajectory has a close genuine trajectory. In 1967 Anosov proved that in a sufficiently small neighbourhood of the hyperbolic set of a diffeomorphism, the shadowing property holds. This result has been called "the shadowing lemma" and has played a key role for the progress of the shadowing theory in dynamical systems. In the talk I will explain the classical shadowing lemma and some similar results obtained later.