Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation

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.