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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The automorphism group of a shift of finite type
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by Mike Boyle, Douglas Lind and Daniel Rudolph PDF
Trans. Amer. Math. Soc. 306 (1988), 71-114 Request permission

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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Additional Information
  • © Copyright 1988 American Mathematical Society
  • 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