In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.

Generic 2D Hamiltonian systems possess partial barriers in their chaotic phase space that restrict classical transport. Quantum mechanically the transport is suppressed if Planck's constant h is large compared to the classical flux, h Φ, such that wave packets and states are localized. In contrast, classical transport is mimicked for h Φ. Designing a quantum map with an isolated partial barrier of controllable flux Φ is the key to investigating the transition from this form of quantum localization to mimicking classical transport. It is observed that quantum transport follows a universal transition curve as a function of the expected scaling parameter Φ/h. We find this curve to be symmetric to Φ/h = 1, having a width of two orders of magnitude in Φ/h, and exhibiting no quantized steps. We establish the relevance of local coupling, improving on previous random matrix models relying on global coupling. It turns out that a phenomenological 2 × 2-model gives an accurate analytical description of the transition curve. In the phase space of generic two-degree-of-freedom (2D) Hamiltonian systems regions of regular and chaotic motion are dynamically separated by impenetrable barriers. Within a chaotic region so-called partial barriers are ubiquitous. They divide it into distinct sub-regions, connected by the turnstile mechanism, which works like a revolving door between two rooms. The volume in phase space, which is transported across the partial barrier in each direction per time is the flux Φ. Partial barriers can originate [1] from a cantorus or the combination of the stable and unstable manifold of a hyperbolic fixed point. A hierarchy of these partial barriers gives rise to a powerlaw decay of correlations and of Poincaré recurrence time distributions [2].