A spectral gap for transfer operators of piecewise expanding maps

Damien Thomine (1)

1 DMA - Département de Mathématiques et Applications - ENS Paris

We provide a simplified proof of the existence, under some assumptions, of a spectral gap for the Perron-Frobenius operator of piecewise uniformly expanding maps on Riemannian manifolds when acting on some Sobolev spaces. Its consequences include, among others, the existence of invariant physical measures, and an exponential decay of correlations for suitable observables. These features are then adapted to different function spaces (functions with bounded variation or bounded oscillation), so as to give a new insight of - and generalize - earlier results.

Mots clés: transfer operator; expanding map; decay of correlation

Domaines: Systèmes dynamiques [math.DS]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. 2010. ⟨hal-00487740⟩