Apparent fractal dimensions in conservative dynamical systems

For conservative dynamical systems, the invariant sets which are in a sense the analog of the strange attractors of dissipative systems, namely the closures of homoclinic orbits of hyperbolic points, are known to have in general integral dimensions. We show however, working numerically on a particular model, that actual numerical estimates for perturbations of an integrable system will necessarily exhibit an apparent fractal dimension, which will be the effective one to all practical purpose. A simple scheme of interpretation is also given.