Abstract
We show that a two-dimensional area-preserving map with Lorentzian potential is a topological horseshoe and uniformly hyperbolic in a certain parameter region. In particular, we closely examine the so-called sector condition, which is known to be a sufficient condition leading to the uniformly hyperbolicity of the system. The map will be suitable for testing the fractal Weyl law as it is ideally chaotic yet free from any discontinuities which necessarily invokes a serious effect in quantum mechanics such as diffraction or nonclassical effects. In addition, the map satisfies a reasonable physical boundary condition at infinity, thus it can be a good model describing the ionization process of atoms and molecules.
Highlights
The periodically kicked one-degree-of-freedom system has been playing and still plays significant roles in the study of chaos in classical and quantum systems
We have provided a sufficient condition for the topological horseshoe and uniform hyperbolicity for the 2-dimensional area-preserving map in which the potential function is expressed by Lorentzian functions
As we mentioned in introduction, our scattering system well fits to the test of the fractal Weyl law conjecture
Summary
The periodically kicked one-degree-of-freedom system has been playing and still plays significant roles in the study of chaos in classical and quantum systems. In a certain parameter regime, the horseshoe is realized and the uniform hyperbolidity was proved to hold [10], and later it has been shown to be true until when the first tangency between stable and unstable manifolds happens [11] Such systems are better suited to examine classical and quantum signatures of ideally chaotic situations and been often taken to be toy models for such a purpose. A complete ideal horseshoe can be formed in the Hénon map but the boundary condition would be physically improper It is, desirable to find a map which has a natural integrable limit, possibly achieved by taking the zero kicking strength limit, and at the same time can become uniformly hyperbolic over a certain parameter space, yet satisfying a physically feasible boundary condition.
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