Sharp Estimates in Ruelle Theorems for Matrix Transfer Operators

A matrix coefficient transfer operator (L\Phi)(x) = P OE(y)\Phi(y), y 2 f \Gamma1 (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 supfh + : 2 Mg and the essential spectral radius by exp supfh + \Gamma r \Delta Ø : 2 Mg: Here M is the set of ergodic f-invariant measures, and for 2 M; h is the measure-theoretic entropy of f , is the largest Lyapunov exponent of the cocycle over f generated by OE, and Ø is the smallest Lyapunov exponent of the differential of f . Keywords. Transfer operator, Lyapunov exponents, multiplicative ergodic theorem, weighted translation operators AMS(MOS) subject classification. 58F20, 58F11, 58F15. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, e-mail: campbelj@hermes.msci.memst.edu. y Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, e-mail: mathyl@mizzou1.m..