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Provides a survey of current research and new approaches in the theory of shadowing of approximate trajectories of dynamical systems Contains novelty approach for proving hyperbolicity by using the sifting method of Liao,  which is both powerful and self-contained  A main feature is the direct, straightforward approach to the results, well written, in an easy-to-read, comfortable pace Can be used right after a course of introduction to dynamical systems or even hyperbolic dynamics