The Automorphism Group of a Shift of Finite Type

Let <!-- MATH $({X_T},{\sigma _T})$ --> be a shift of finite type, and <!-- MATH $G = \operatorname{aut} ({\sigma _T})$ --> denote the group of homeomorphisms of commuting with <!-- MATH ${\sigma _T}$ --> . We investigate the algebraic properties of the countable group and the dynamics of its action on and associated spaces. Using "marker" constructions, we show contains many groups, such as the free group on two generators. However, 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 coordinates leads to a new and nontrivial topological invariant of <!-- MATH ${\sigma _T}$ --> whose exact value is not known. We prove that, modulo a few points of low period, acts transitively on the set of points with least <!-- MATH ${\sigma _T}$ --> -period . Using -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 on the dimension group of <!-- MATH ${\sigma _T}$ --> is investigated. We show there are no proper infinite compact -invariant sets. We give a complete characterization of the -orbit closure of a continuous probability measure, and deduce that the only continuous -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.