Periodic Points and Automorphisms of the Shift

The automorphism group of a topological Markov shift is studied by way of periodic points and unstable sets. A new invariant for automorphisms of dynamical systems, the gyration function, is used to characterize those automorphisms of finite subsystems of the full shift on $n$ symbols which can be extended to a composition of involutions of the shift. It is found that for any automorphism $U$ of a subshift of finite type $S$, for all large integers $M$ the map $U{S^M}$ is a topological Markov shift whose unstable sets equal those of $S$. This fact yields, by way of canonical measures and dimension groups, information about dynamical properties of $U{S^k}$ such as the zeta function and entropy.