 95406 Campbell J., Latushkin Y.
 Sharp Estimates in Ruelle Theorems for
Matrix Transfer Operators
(55K, LaTeX)
Sep 1, 95

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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
\[\exp\sup\{h_\nu+\lambda_\nur\cdot\chi_\nu:\nu\in\calM\}.\]
Here $\calM$ is the set of ergodic $f$invariant measures, and for
$\nu \in {\cal M}, \; h_{\nu}$
is the measuretheoretic 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 $f$.
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