Fractal Weyl law for skew extensions of expanding maps

Jean-François Arnoldi (1)

1 IF - Institut Fourier

Abstract. We consider compact Lie groups extensions of expanding maps of the circle, essentially restricting to \mbox{U}(1) and \mbox{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 \left\{ \hat{F}_{\alpha}\right\} when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues (the Ruelle resonances) of these operators out of some fixed disc centered on 0 in the complex plane.

Mots clés: Compact Lie group extensions; Partially hyperbolic systems; Ruelle resonances; Decay of correlations; Fractal Weyl law; Semiclassical analysis

Domaines: Systèmes dynamiques [math.DS]

Jean-François Arnoldi. Fractal Weyl law for skew extensions of expanding maps. 2011. ⟨hal-00654993⟩