J. Campbell et Y. Latushkin, SHARP ESTIMATES IN RUELLE THEOREMS FOR MATRIX TRANSFER OPERATORS, Communications in Mathematical Physics, 185(2), 1997, pp. 379-396
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 s
pectral radius of L is estimated by exp (sup{h(v) + lambda(v) : v Sigm
a 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 mea
sures, 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 gen
erated by phi, and chi(v) is the smallest Lyapunov exponent of the dif
ferential of f.