A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Holder spaces

We study Ruelle's transfer operator C induced by a Cr+1-smooth expanding map p of a smooth manifold and a C-r-smooth bundle automorphism Phi of a real vector bundle S. We prove the following exact formula for the essential spectral radius of L on the space C-r,C-alpha of r-times continuously differentiable sections of E with alpha-Holder rth derivative: [GRAPHICS] where Erg is the set of psi-ergodic measures, h(v) the entropy of psi with respect to v, lambda(v) the largest Lyapunov exponent of the cocycle induced by Phi, and X-v the smallest Lyapunov exponent for the differential Dpsi.