Campbell J., Latushkin Y.
Sharp Estimates in Ruelle Theorems for
Matrix Transfer Operators
(55K, LaTeX)

ABSTRACT.  A matrix coefficient transfer
operator $(\calL\Phi)(x)=\sum\phi(y)\Phi(y)$,
$y\in f^{-1}(x)$ on the space of $C^r$-sections of an
$m$-dimensional
vector bundle over $n$-dimensional compact manifold is considered.
The spectral radius of $\calL$
is estimated by \newline $\displaystyle{\exp \sup \{ h_\nu +
\lambda_\nu:\nu\in\calM\}}$
and the essential spectral radius by
\[\exp\sup\{h_\nu+\lambda_\nu-r\cdot\chi_\nu:\nu\in\calM\}.\]
Here $\calM$ is the set of ergodic $f$-invariant measures, and for
$\nu \in {\cal M}, \; h_{\nu}$
is the measure-theoretic entropy of $f$, $\lambda_\nu$  is
the largest Lyapunov exponent of the cocycle over $f$
generated by $\phi$, and $\chi_\nu$ is the
smallest Lyapunov exponent
of the differential of $f$.