Shadowing property: State of art and perspectives

We consider a dynamical system generated by a diffeomorphism or a vector field. The shadowing problem is related to the following question: Under which conditions, for any pseudotrajectory of a dynamical system there exists a close trajectory? This notion is important for stability theory and for theoretical motivation of numerical simulations.

It is well known that a dynamical system has shadowing property in a neighborhood of a hyperbolic set. In fact it is informally believed that almost only hyperbolic systems have shadowing. While construction of non-hyperbolic example with shadowing is easy, converting this informal feeling to rigorous statements is more tricky.

I give an overview of main results in this theory and describe perspectives for future research. In particular problems relating shadowing to skew products and stochastic stability.