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

Scott Schmieding

doi:10.1017/etds.2019.9

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

Scott Schmieding

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Let be a shift of finite type and its corresponding automorphism group. Associated to are certain Lyapunov exponents, which describe asymptotic behavior of the sequence of coding ranges of. We give lower bounds on in terms of the spectral radius of the corresponding action of on the dimension group associated to. We also give lower bounds on the topological entropy in terms of a distinguished part of the spectrum of the action of on the dimension group, but show that, in general, is not bounded below by the logarithm of the spectral radius of the action of on the dimension group.

Original language	English (US)
Pages (from-to)	2552-2570
Number of pages	19
Journal	Ergodic Theory and Dynamical Systems
Volume	40
Issue number	9
DOIs	https://doi.org/10.1017/etds.2019.9
State	Published - Sep 1 2020

Keywords

Lyapunov exponents
automorphism
entropy
shift of finite type

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1017/etds.2019.9

Cite this

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

abstract = "Let be a shift of finite type and its corresponding automorphism group. Associated to are certain Lyapunov exponents, which describe asymptotic behavior of the sequence of coding ranges of. We give lower bounds on in terms of the spectral radius of the corresponding action of on the dimension group associated to. We also give lower bounds on the topological entropy in terms of a distinguished part of the spectrum of the action of on the dimension group, but show that, in general, is not bounded below by the logarithm of the spectral radius of the action of on the dimension group.",

keywords = "Lyapunov exponents, automorphism, entropy, shift of finite type",

author = "Scott Schmieding",

note = "Funding Information: Acknowledgement. This research was supported in part by the National Science Foundation grant {\textquoteleft}RTG: Analysis on manifolds{\textquoteright} at Northwestern University. I would like to thank Ville Salo for communicating to us the example in Remark 4.8, and Masakazu Nasu for referring me to results contained in his article [22]. I am especially grateful to Mike Boyle for many helpful discussions and comments. Publisher Copyright: {\textcopyright} Cambridge University Press, 2019.",

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N1 - Funding Information: Acknowledgement. This research was supported in part by the National Science Foundation grant ‘RTG: Analysis on manifolds’ at Northwestern University. I would like to thank Ville Salo for communicating to us the example in Remark 4.8, and Masakazu Nasu for referring me to results contained in his article [22]. I am especially grateful to Mike Boyle for many helpful discussions and comments. Publisher Copyright: © Cambridge University Press, 2019.

PY - 2020/9/1

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N2 - Let be a shift of finite type and its corresponding automorphism group. Associated to are certain Lyapunov exponents, which describe asymptotic behavior of the sequence of coding ranges of. We give lower bounds on in terms of the spectral radius of the corresponding action of on the dimension group associated to. We also give lower bounds on the topological entropy in terms of a distinguished part of the spectrum of the action of on the dimension group, but show that, in general, is not bounded below by the logarithm of the spectral radius of the action of on the dimension group.

AB - Let be a shift of finite type and its corresponding automorphism group. Associated to are certain Lyapunov exponents, which describe asymptotic behavior of the sequence of coding ranges of. We give lower bounds on in terms of the spectral radius of the corresponding action of on the dimension group associated to. We also give lower bounds on the topological entropy in terms of a distinguished part of the spectrum of the action of on the dimension group, but show that, in general, is not bounded below by the logarithm of the spectral radius of the action of on the dimension group.

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KW - automorphism

KW - entropy

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

Abstract

Keywords

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