Fractal Weyl law for skew extensions of expanding maps

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 $\widehat{F}$. Harmonic analysis yields a natural decomposition $\widehat{F}=\oplus {\widehat{F}}_{\alpha }$, 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 ${\widehat{F}}_{\alpha }$ 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.