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The Automorphism Group of a Shift of Finite Type
| Content Provider | Scilit |
|---|---|
| Author | Boyle, Mike Lind, Douglas Rudolph, Daniel |
| Copyright Year | 1988 |
| Description | 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. |
| Related Links | https://www.ams.org/tran/1988-306-01/S0002-9947-1988-0927684-2/S0002-9947-1988-0927684-2.pdf |
| ISSN | 00029947 |
| e-ISSN | 10886850 |
| DOI | 10.2307/2000831 |
| Journal | Transactions of the American Mathematical Society |
| Issue Number | 1 |
| Volume Number | 306 |
| Language | English |
| Publisher | Duke University Press |
| Publisher Date | 1988-03-01 |
| Access Restriction | Open |
| Subject Keyword | Logic Mathematical Physics Shift Sigma X_t Finite Type Survey Function Infinite Invariant Automorphism |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics |
