Focusing on the theory of shadowing of approximate trajectories (pseudotrajectories) of
dynamical systems this book surveys recent progress in establishing relations between
shadowing and such basic notions from the classical theory of structural stability as
hyperbolicity and transversality. Special attention is given to the study of quantitative
shadowing properties such as Lipschitz shadowing (it is shown that this property is equivalent
to structural stability both for diffeomorphisms and smooth flows) and to the passage to
robust shadowing (which is also equivalent to structural stability in the case of
diffeomorphisms while the situation becomes more complicated in the case of flows). Relations
between the shadowing property of diffeomorphisms on their chain transitive sets and the
hyperbolicity of such sets are also described. The book will allow young researchers in the
field of dynamical systems to gain a better understanding of new ideas in the global
qualitative theory. It will also be of interest to specialists in dynamical systems and their
applications.
