Do Chaotic Trajectories Care About Self-Similarity?

We investigate the relation between the chaotic dynamics and the hierarchical phase-space structure of generic Hamiltonian systems. We demonstrate that even in ideal situations when the phase space is dominated by an exactly self-similar structure, the long-time dynamics is {\it not} dominated by this structure. This has consequences for the power-law decay of correlations and Poincar\'e recurrences.