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Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation
Published online by Cambridge University Press: 25 March 2019
Abstract
Let $(X_{A},\unicode[STIX]{x1D70E}_{A})$ be a shift of finite type and
$\text{Aut}(\unicode[STIX]{x1D70E}_{A})$ its corresponding automorphism group. Associated to
$\unicode[STIX]{x1D719}\in \text{Aut}(\unicode[STIX]{x1D70E}_{A})$ are certain Lyapunov exponents
$\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$, which describe asymptotic behavior of the sequence of coding ranges of
$\unicode[STIX]{x1D719}^{n}$. We give lower bounds on
$\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$ in terms of the spectral radius of the corresponding action of
$\unicode[STIX]{x1D719}$ on the dimension group associated to
$(X_{A},\unicode[STIX]{x1D70E}_{A})$. We also give lower bounds on the topological entropy
$h_{\text{top}}(\unicode[STIX]{x1D719})$ in terms of a distinguished part of the spectrum of the action of
$\unicode[STIX]{x1D719}$ on the dimension group, but show that, in general,
$h_{\text{top}}(\unicode[STIX]{x1D719})$ is not bounded below by the logarithm of the spectral radius of the action of
$\unicode[STIX]{x1D719}$ on the dimension group.
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- Original Article
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- © Cambridge University Press, 2019
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