Fractal Weyl law for skew extensions of expanding maps

TL;DR

This paper establishes a fractal Weyl law for Ruelle resonances associated with skew extensions of expanding maps on the circle, revealing spectral properties linked to the underlying fractal structures.

Contribution

It extends previous work by proving a fractal Weyl upper bound for eigenvalues of transfer operators in Lie group extensions of expanding maps.

Findings

Proves a spectral gap for transfer operators using semiclassical analysis.

Establishes a fractal Weyl upper bound for eigenvalues of Ruelle resonances.

Connects spectral properties to fractal geometry of the underlying dynamical system.

Abstract

We consider compact Lie groups extensions of expanding maps of the circle, essentially restricting to U(1) and SU(2) extensions. The central object of the paper is the associated Ruelle transfer (or pull-back) operator $\hat{F}$. Harmonic analysis yields a natural decomposition $\hat{F}=\oplus\hat{F}_{\alpha}$, where $\alpha$ indexes the irreducible representation spaces. Using Semiclassical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family ${\hat{F}_{\alpha}}$ when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues of these operators (the Ruelle resonances) out of some fixed disc centered on 0 in the complex plane.