Sharp estimates in Ruelle theorems for matrix transfer operators


A matrix coefficient transfer operator (script L sign Φ)(x) = Σ θ(y)Φ(y), y ∈ f-1(x) on the space of Cr -sections of an m-dimensional vector bundle over n-dimensional compact manifold is considered. The spectral radius of script L sign is estimated by exp (supj{hv + λv : v ∈ M }) and the essential spectral radius by exp (sup{hv + λv - r · λv : v ∈ M}) · Here M is the set of ergodic f-invariant measures, and for v ∈ M, hv is the measuretheoretic entropy of f, λv is the largest Lyapunov exponent of the cocycle over f generated by Φ, and Xv is the smallest Lyapunov exponent of the differential of f.

Publication Title

Communications in Mathematical Physics