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

TL;DR

This paper investigates the automorphisms of shifts of finite type, establishing bounds on Lyapunov exponents and entropy based on spectral properties of their actions on the associated dimension group, revealing nuanced relationships between these invariants.

Contribution

It provides new lower bounds for Lyapunov exponents and entropy of automorphisms in terms of spectral data from the dimension group, advancing understanding of their dynamical complexity.

Findings

Lower bounds on Lyapunov exponents in terms of spectral radius.

Lower bounds on topological entropy related to spectrum of the dimension group.

Entropy is not always bounded below by the logarithm of the spectral radius.

Abstract

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.