Lipschitz shadowing implies structural stability

We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability. As a corollary, we show that an expansive diffeomorphism having the Lipschitz shadowing property is Anosov.Comment: 11 page

Convergence of discretized attractors for parabolic equations on the line

Abstract We show that, for a semilinear parabolic equation on the real line satisfying a dissipativity condition, global attractors of time-space discretizations converge (with respect to the Hausdorff semi-distance) to the attractor of the continuous system as the discretization steps tend to zero. The attractors considered correspond to pairs of function spaces (in the sense of Babin-Vishik) with weighted and locally uniform norms (taken from Mielke-Schneider) used for both the continuous and the discrete system

Complete Families of Pseudotrajectories and Shape of Attractors

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