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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
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