The Automorphism Group of a Shift of Finite Type
نویسندگان
چکیده
Let (Xt,ot) be a shift of finite type, and G = aut(or) denote the group of homeomorphisms of Xt commuting with ctWe investigate the algebraic properties of the countable group G and the dynamics of its action on Xt 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 ot 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 or-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 o~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. Table of
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