Le rayon spectral essentiel de l'opérateur de Ruelle sur des variétés höldériennes

Abstract

We study the deterministic and random Ruelle transfer operator L induced by an expanding map ƒ of a smooth n-dimensional manifold X and a bundle automorphism ϑ of a rank L vector bundle E. We prove the following exact formula for the essential spectral radius of L on the space C r, α of r-times continuously differentiable sections of E with α-H o ̈ lder r-th derivative: r ess( L;C r,α) = exp ( vϵrg sup {h v + λ v, − (r+α)x v}) , where Erg denotes the set of ƒ-ergodic measures, h v the entropy of ƒ with respect to v, λ v the largest Lyapunov exponent of the cocycle ϑ (x) = ϑ(ƒ k−1(x) ·…· ϑ(x) , and x v the smallest Lyapunov exponent of the differential Dƒ k(x) , x ϵ X, k = 1, 2,…. A similar result holds for the random case.