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 $\varphi \in \text{Aut}({\sigma }_A)$ are certain Lyapunov exponents ${\alpha }^{-}(\varphi ),{\alpha }^{+}(\varphi )$ which describe asymptotic behavior of the sequence of coding ranges of ${\varphi }^n$. We give lower bounds on ${\alpha }^{-}(\varphi ),{\alpha }^{+}(\varphi )$ in terms of the spectral radius of the corresponding action of $\varphi$ on the dimension group associated to $(X_A,{\sigma }_A)$. We also give lower bounds on the topological entropy $h_{top}(\varphi )$ in terms of a distinguished part of the spectrum of the action of $\varphi$ on the dimension group, but show that in general $h_{top}(\varphi )$ is not bounded below by the logarithm of the spectral radius of the action of $\varphi$ on the dimension group.