Although a numerically computed Chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (``shadows'') the numerically computed one. Therefore, the Fractal structure of chaotic trajectories seen in computer maps is real.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 18-19, 1993.