Ulam method and fractal Weyl law for Perron-Frobenius operators

@article{Ermann2010UlamMA,
  title={Ulam method and fractal Weyl law for Perron-Frobenius operators},
  author={Leonardo Ermann and Dima L. Shepelyansky},
  journal={The European Physical Journal B},
  year={2010},
  volume={75},
  pages={299-304}
}
We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we… 
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References

SHOWING 1-10 OF 42 REFERENCES
Resonant eigenstates for a quantized chaotic system
We study the spectrum of quantized open maps as a model for the resonance spectrum of quantum scattering systems. We are particularly interested in open maps admitting a fractal repeller. Using the…
Fractal Weyl law for quantum fractal eigenstates.
  • D. Shepelyansky
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2008
The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is…
Ruelle?Perron?Frobenius spectrum for Anosov maps
We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows…
Distribution of Resonances for Open Quantum Maps
We analyze a simple model of quantum chaotic scattering system, namely the quantized open baker’s map. This model provides a numerical confirmation of the fractal Weyl law for the semiclassical…
Rigorous numerical approximation of Ruelle?Perron?Frobenius operators and topological pressure of expanding maps
It is well known that for different classes of transformations, including the class of piecewise C2 expanding maps T : [0, 1] ↺, Ulam's method is an efficient way to numerically approximate the…
Fractal Weyl laws for chaotic open systems.
TLDR
The conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller is presented and the result generalizes the Weyl law for thedensity of states of a closed system to chaotic open systems.
Scaling in multifractals: Discretization of an eigenvalue problem.
  • Kovács, Tél
  • Mathematics, Physics
    Physical review. A, General physics
  • 1989
A matrix approximant to the generalized Frobenius-Perron equation is presented, the largest eigenvalues of which approach the most important eigenvalues of the operator (e.g., the so-called free…
The Selberg Zeta Function for Convex Co-Compact Schottky Groups
We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍn+1: in strips parallel to the imaginary axis the zeta function is…
Semiclassical structure of chaotic resonance eigenfunctions.
TLDR
The limit of a sequence of eigenstates [psi(variant Planck's over)] 2pi-->0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate.
Distribution of resonances in the quantum open baker map.
TLDR
This work verified that the validity of the fractal Weyl law holds in all cases and introduces new ingredients to the association among the classical escape and quantum decay rates.
...
1
2
3
4
5
...