  Let $(X_{A},\sigma_{A})$ be a shift of finite type and
$\text{Aut}(\sigma_{A})$ its corresponding automorphism group. Associated to
$\phi \in \text{Aut}(\sigma_{A})$ are certain Lyapunov exponents
$\alpha^{-}(\phi), \alpha^{+}(\phi)$ which describe asymptotic behavior of the
sequence of coding ranges of $\phi^{n}$. We give lower bounds on
$\alpha^{-}(\phi), \alpha^{+}(\phi)$ in terms of the spectral radius of the
corresponding action of $\phi$ on the dimension group associated to
$(X_{A},\sigma_{A})$. We also give lower bounds on the topological entropy
$h_{top}(\phi)$ in terms of a distinguished part of the spectrum of the action
of $\phi$ on the dimension group, but show that in general $h_{top}(\phi)$ is
not bounded below by the logarithm of the spectral radius of the action of
$\phi$ on the dimension group.
