Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems

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.

Figure 1

(a) Phase space of the standard map at $\kappa =2.9$ with regular (thin solid gray lines) and chaotic (gray points) orbits, three partial barriers (thick solid colored lines) and the preimage of the outermost partial barrier (dotted magenta line). (b) Chaotic saddle of the opened map (gray-shaded absorbing stripes) colored according to the regions ($A_0$: light green, $A_1$: dark blue). (c) Rescaled hierarchical fractal Weyl laws ${\tilde{\mathcal{N}}}_j$ vs $h^{-1}$ (filled symbols) counting hierarchical resonance states in the outer ($A_0$, triangles) and inner ($A_1$, circles) chaotic regions (with corresponding typical Husimi representations for $h=1/1000$). Their power-law scaling is compared to the rescaled box-counting scaling ${\tilde{N}}_j^{\mathrm{bc}}$ vs ${\epsilon }^{-2}$ (open symbols) with fractal dimension ${\delta }_j$ in region $A_j$ of the chaotic saddle.

Figure 2

Distributions $P(\gamma )$ of decay rates $\gamma$ for hierarchical resonance states of the standard map at $\kappa =2.9$ located in regions $A_0$ (right, yellow) and $A_1$ (left, red) for $1/h=12800$ and corresponding Husimi representations of typical states. Short-lived states ($\gamma >{\gamma }_{\mathrm{c}}$) are not counted in the fractal Weyl law.

Figure 3

Rescaled hierarchical fractal Weyl laws ${\tilde{\mathcal{N}}}_j$ vs $h^{-1}$ (filled symbols) counting hierarchical resonance states in the outer ($A_0$, triangles), central ($A_1$, circles), and inner ($A_2$, squares) chaotic regions for the hierarchical model system, and corresponding typical Husimi representations (right) for $h=1/1115$. Comparison to rescaled box-counting scaling ${\tilde{N}}_j^{\mathrm{bc}}$ vs ${\epsilon }^{-2}$ (open symbols) of the chaotic saddle (inset) with fractal dimension ${\delta }_j$ in region $A_j$.