Can simple renormalization theories describe the trapping of chaotic trajectories in mixed systems?

We investigate the relation between the chaotic dynamics and the hierarchical phase-space structure of the standard map as an example for generic Hamiltonian systems with a mixed phase space. We demonstrate that even in ideal situations when the phase-space structure is dominated by a single scaling, the long-time dynamics is not dominated by this scaling. This has consequences for the power-law decay of correlations and Poincaré recurrences.