Weyl law for fat fractals

TL;DR

This paper investigates the quantum properties of certain piecewise linear maps with fat fractal structures, demonstrating a modified fractal Weyl law that relates quantum state localization to the fractal's exterior dimension.

Contribution

It introduces a quantum analysis of fat fractals in piecewise linear maps, establishing a modified Weyl law linking quantum states to fractal geometry.

Findings

Fraction of localized states follows a modified fractal Weyl law

Exponent of the law equals the exterior dimension of the fat fractal

Supports conjecture of fat fractal structures in dynamical systems

Abstract

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 exterior dimension of the fat fractal.