Weyl law for fat fractals

  title={Weyl law for fat fractals},
  author={Mar{\'i}a Elena Spina and Ignacio Garc{\'i}a-Mata and Marcos Saraceno},
  journal={arXiv: Chaotic Dynamics},
It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the… Expand

Figures and Tables from this paper

On a Variation of the Definition of Limit: Some Analytic Consequences
The basic formalism of a novel scale invarinat nonlinear analysis is presented. A few analytic number theoretic results are derived independent of standard approaches.


Fractal Weyl law for quantum fractal eigenstates.
  • D. Shepelyansky
  • Mathematics, Medicine
  • 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 isExpand
Fractal Weyl laws for chaotic open systems.
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. Expand
Fractal Weyl laws in discrete models of chaotic scattering
We analyse simple models of quantum chaotic scattering, namely quantized open baker's maps. We numerically compute the density of quantum resonances in the semiclassical regime. This densityExpand
Lossless Digital Filter Overflow Oscillations; Approximation of Invariant Fractals
We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. ForExpand
Elliptic behaviour in the sawtooth standard map
Abstract This paper examines the standard map with sawtooth nonlinearity when the eigenvalues of the Jacobian lie on the unit circle. This is an area-preserving map of the torus to itself that isExpand
Weyl's law and quantum ergodicity for maps with divided phase space
For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for theExpand
Fractal Weyl laws for quantum decay in dynamical systems with a mixed phase space.
  • M. Kopp, H. Schomerus
  • Physics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
It is shown that resonances in open quantum systems can follow a modified fractal Weyl law, even when their classical dynamics is not globally chaotic but also contains domains of regular motion, and connected to emerging quantum-to-classical correspondence. Expand
On the quantum cat and sawtooth maps—Return to generic behaviour
Abstract The quantization of the continuous cat maps on the torus has led to rather pathological quantum objects [1]. The non-generic behaviour of this model has led some to conclude that theExpand
Sensitive dependence on parameters in nonlinear dynamics.
  • Farmer
  • Physics, Medicine
  • Physical review letters
  • 1985
Two qualitatively different types of dynamical behavior can be so tightly interwoven that it becomes impossible predict when a small change in parameters will cause a change in qualitativeExpand
Hamiltonian mappings and circle packing phase spaces
We introduce three area preserving maps with phase space structures which resemble circle packings. Each mapping is derived from a kicked Hamiltonian system with one of the three different phaseExpand