The automorphism group of a shift of finite type

Boyle, Mike; Lind, Douglas; Rudolph, Daniel

doi:10.1090/S0002-9947-1988-0927684-2

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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