Hyperbolic trajectories of time discretizations

The problem of persistence of trajectories of time discretizations is addressed. Analogues of essential concepts in the theory of hyperbolic flows are formulated for time discretizations. We show these are appropriate by proving that the underlying flow inherits a hyperbolic trajectory of the time discretization. Approximation properties and inherited hyperbolicity properties are analyzed. Here explicit estimates in terms of the key problem parameters are provided. An auxiliary result on roughness of exponential dichotomies for time discretizations may be of independent interest. Directions of future research are indicated. (C) 2004 Elsevier Ltd. All rights reserved.