Sharp estimates in Ruelle theorems for matrix transfer operators

A matrix coefficient transfer operator (L Phi)(x) = Sigma phi(y)Phi(y), y epsilon f(-1)(x) on the space of C-r-sections of an m-dimensional vector bundle over n-dimensional compact manifold is considered. The spectral radius of L is estimated by exp (sup{h(v) + lambda(v) : v Sigma M}) and the essential spectral radius by exp (sup{h(v) + lambda(v) - r.chi(v): v epsilon M)). Here M is the set of ergodic f-invariant measures, and for v epsilon M, h(v) is the measure theoretic entropy of f, lambda(v) is the largest Lyapunov exponent of the cocycle over f generated by phi, and chi(v) is the smallest Lyapunov exponent of the differential of f.