THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

Van Cyr; Bryna Kra

doi:10.1017/fms.2015.3

THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

Van Cyr, Bryna Kra

Mathematics

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

20 Scopus citations

Abstract

For a finite alphabet A and shift X ⊆ A^Z whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group Aut(X). For such systems, we show that every finitely generated subgroup of Aut(X) is virtually ℤ^d, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if X is transitive, then Aut(X) is virtually ℤ; if X has dense aperiodic points, then Aut(X) is virtually ℤ^d. We also classify all finite groups that arise as the automorphism group of a shift.

Original language	English (US)
Article number	e5
Journal	Forum of Mathematics, Sigma
Volume	3
DOIs	https://doi.org/10.1017/fms.2015.3
State	Published - Jan 1 2015

ASJC Scopus subject areas

Analysis
Theoretical Computer Science
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Mathematics

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10.1017/fms.2015.3

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title = "THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY",

abstract = "For a finite alphabet A and shift X ⊆ AZ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group Aut(X). For such systems, we show that every finitely generated subgroup of Aut(X) is virtually ℤd, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if X is transitive, then Aut(X) is virtually ℤ; if X has dense aperiodic points, then Aut(X) is virtually ℤd. We also classify all finite groups that arise as the automorphism group of a shift.",

author = "Van Cyr and Bryna Kra",

note = "Funding Information: We thank J. Davis for pointing us to reference [15], P. Brooksbank for helpful conversations, and the referee for numerous helpful comments. The second author was partially supported by NSF grant DMS-1200971 Publisher Copyright: {\textcopyright} 2015 The Author(s).",

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THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

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