We study the behavior of families of δ-trajectories (pseudotrajectories) near an attractor of a finite-dimensional dynamical system. It is shown that for arbitrary dynamical system the boundary of any attractor can be uniformly approximated by points of pseudotrajectories beginning at points of a parametrizing set. This result is refined for C^0-generic systems and for structurally stable diffeomorphisms. Special Lyapunov functions are used to estimate the rate of approximation in the case of a structurally stable diffeomorphism.Supported in part by RFFI, grant G94-1.3.90
