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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The automorphism group of a shift of finite type


Authors: Mike Boyle, Douglas Lind and Daniel Rudolph
Journal: Trans. Amer. Math. Soc. 306 (1988), 71-114
MSC: Primary 54H20; Secondary 20B27, 28D15, 34C35, 58F11
DOI: https://doi.org/10.1090/S0002-9947-1988-0927684-2
MathSciNet review: 927684
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Abstract: Let $({X_T},{\sigma _T})$ be a shift of finite type, and $G = \operatorname {aut} ({\sigma _T})$ denote the group of homeomorphisms of ${X_T}$ commuting with ${\sigma _T}$. We investigate the algebraic properties of the countable group $G$ and the dynamics of its action on ${X_T}$ and associated spaces. Using "marker" constructions, we show $G$ contains many groups, such as the free group on two generators. However, $G$ is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on $n$ coordinates leads to a new and nontrivial topological invariant of ${\sigma _T}$ whose exact value is not known. We prove that, modulo a few points of low period, $G$ acts transitively on the set of points with least ${\sigma _T}$-period $n$. Using $p$-adic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of $G$ on the dimension group of ${\sigma _T}$ is investigated. We show there are no proper infinite compact $G$-invariant sets. We give a complete characterization of the $G$-orbit closure of a continuous probability measure, and deduce that the only continuous $G$-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.


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Keywords: Shift of finite type, symbolic dynamics, automorphism group, dimension group, invariant measure, residual finiteness, gyration function, periodic points
Article copyright: © Copyright 1988 American Mathematical Society